DISCLAIMER: I do not own JoJo's Bizarre Adventure or Superman, Justice League or DC/Vertigo Comics.
Dio sat in his darkened tower, reading his copy of the Bible and scoffing at every single message he saw, something he had already done with plenty of other religious and historical documents and even the copy of J.R.R. Tolkien's The Lord of the Rings and the copy of C.S. Lewis' The Chronicles of Narnia he had read. Though he appreciated the literature he scoffed at the idea of beliefs and moral codes.
"What nonsense. The great Dio is the only god, the only messiah, the be all and end all of ideologies and leaders! The messages in these writings are all useless, useless, useless, useless, useless…"
He had planned to say the word "useless" at least a few more time, and at the low volume he was speaking at only he would have heard it, but suddenly an explosion outside shook his tower, filling it with cracks. Dio closed his book, standing up and calling for his attendant. The response to his call was the top of his tower being blasted off, and had it been anything even resembling daytime Dio would have died right there, but his history of good luck had held and the Sun had gone completely down several hours before, and the time was well past midnight. Even Dio wasn't arrogant enough to not feel slightly afraid knowing that even the smallest amount of direct sunlight would kill him.
Doomsday stood up, turning to face the oncoming Supergirl and punching her hard in the face, knocking her straight onto her back, her face terribly bruised. Doomsday reached down and grabbed her by the face, not even reacting as she fired her Heat Vision into his clawed hand, causing the stink of burned flesh to rise up in a thin collumn of smoke. Supergirl continued firing until another punch to her face knocked her out with a sharp snap which was the result of her neck breaking. The strike also shook the entire Earth on its axis, knocking Aquaman off his throne at the bottom of the sea and scaring Suzi Q in Japan. A defeated Supergirl went limp under Doomsday's hand, and as Doomsday went to crush her, a deep voice yelled out a loud statement. Doomsday recognised the word kill and a strange laugh, but didn't understand the rest of it.
Doomsday turned to face the newcomer, and saw a man dressed almost entirely in gold with golden hair. A smarter creature may have noticed Dio's stance or the cocky look on his face, but all Doomsday saw was a gold man. Dio started to speak about his superiority at his usual loud volume, but Doomsday lunged at him, and Dio instantly became serious, sending The World out to attack Doomsday, the golden Stand punching the creature repeatedly, knocking it in a few directions. The World punched Doomsday literally dozens of times from every direction, and finally Doomsday tripped backwards due to his balance being upset, landing heavily on his back and roaring unintelligently. Doomsday was on his feet in an instant, and lunged at Dio full force. As Doomsday closed in on Dio, time suddenly froze, and Doomsday simply stopped in midair. Dio hovered over to Doomsday, smirking arrogantly.
Dio slowly circled around the frozen Doomsday, and slowly counted the seconds ticking away.
"Feel the power of the great Dio! You can't do anything! This is my World of stopped time! You won't even know what killed you or when! Useless, useless, useless, useless, useless, useless, useless, useless, useless, useless…"
Dio's verbal diarrhoea continued until his time stop ceased, and Doomsday flew past him, slamming into the ground where Dio had been and sending cracks through the entire city, even causing an earthquake in the United States. Doomsday rounded on Dio and lunged again, but Dio flew upwards, leaving Doomsday grasping at empty air.
Dio hovered several metres out of Doomsday's reach, gloating and celebrating his superiority, when Doomsday landed on the ground and instantly jumped back up at Dio, reaching him and swinging his fist at Dio, who had been so busy talking about his superiority that he was only just in time to activate his time stop, freezing Doomsday in midair. Dio glared at the frozen face of the creature, who was right in front of him and would have hit him had Dio been an instant slower.
"How dare you? How dare you seek to lay hands on the great Dio?"
Dio continued to talk about his own superiority for the remaining few seconds before making The World punch Doomsday repeatedly, until time resumed and Doomsday continued flying, going past Dio and up into the air, before coming back down and swinging his claw at Dio who again stopped time but now instead of wasting his time Dio threw The World at Doomsday, the golden Stand punching Doomsday continuously for the entire eleven seconds in which time was stopped. When time resumed, Dio hovered back just as Doomsday continued falling, but was confused at how the creature was still flailing and trying to hit him when any other creature would have died from all the punches The World had landed on it.
Doomsday crashed into the ground, and jumped up again, only for Dio to again freeze him and attack with The World, though when time resumed Doomsday had suffered only minimal damage. The unintelligent creature dropped when time resumed, and Dio flew away, realizing he had to think of a new way to attack it. As Doomsday jumped up at Dio again, the vampire stopped time again, this time gripping Doomsday by the face and dropping his own body temperature, and ability he was glad he had regained though he had doubted he ever would. Dio then hovered back and fired his Space Ripper Stingy Eyes at Doomsday, attempting to cut him in multiple places, but when time resumed the ice instantly shattered and the jets of fluid hadn't even come close to cutting Doomsday.
Dio quickly flew higher to avoid Doomsday's charge, and was starting to feel as if he may actually be at a disadvantage here. After considering it for a second he dismissed it, deciding that Doomsday was an animal like any other and Dio would find a way to kill him. Dio then stopped time and threw The World at Doomsday again, watching as the gold Stand stuck the creature with its fists and with blades, but to no effect other than the blades breaking. When time resumed Dio had already started running away. It had just occurred to him that this simple creature may not be able to swim due to its density, and if he dropped it in the ocean or something similar he could stop it.
Doomsday lunged at the man in gold again, not interpreting his constant vanishing acts at all, and not even possessing the intelligence to register the punches that kept hitting him, but rather just finding the man in gold and lunging at him. Dio recognised this pattern but as far as he was concerned an ordinary human was exactly the same, so he simply stayed ahead and headed for the nearest body of water, which was very close by as it happened. Dio hovered over the water and Doomsday lunged at him, missing and falling into the water. Dio waited to see if Doomsday would sink, but the creature instantly jumped out, completely unhindered by the water.
Dio dodged the oncoming creature, watching as it slammed into a building and smashed the building into oblivion. The creature recovered instantly and again jumped at Dio, who again froze time and punched it repeatedly to the same lack of effect. After repeating this process an almost incalculable number of times Dio realized it was almost morning, and even the slightest dose of sunlight would kill him. Dio instantly froze time and ran as far away from Doomsday as he could in the eleven seconds of frozen time, hiding himself in a building's basement and barricading the door with a large set of shelves.
As time resumed Doomsday shattered yet another building, turning to look for Dio but unable to find him. Doomsday sniffed the air and charged in the direction he thought Dio had gone, shattering buildings that happened to be in his path. Doomsday soon lost the scent of Dio as it was in several places, and simply started shattering the buildings that he was near, which happened to be the buildings around the one Dio was in. Various townspeople fled as the buildings were shattered but Doomsday didn't even notice them even though they were screaming.
The Sun came up as Doomsday shattered every building he came close to, before he finally smashed the right one, exposing the basement Dio was in and even smashing its walls. Dio cried out as the weak morning sunlight streamed into the basement, and the vampire instantly turned to dust and ashes, not even leaving his clothes behind. Doomsday was too stupid to understand the cause for the demise of its enemy or for that matter even notice it had happened, and attacked the spot where Dio had been, completely destroying the empty basement.
Analysis:
Apologies in advance for how short this chapter is; there really wasn't much either party could do beyond the same basic moves which were going to go nowhere.
The reason for this outcome is that on one hand Dio and The World/Za Warudo/Za Waldo obviously don't have the attack power to destroy Doomsday who to date even Superman has been unable to kill, at least not permanently. Any damage done by The World's fists will regenerate very quickly due to how Doomsday can regenerate from almost anything. While major injuries like decapitation take a lot longer, the minor injuries The World could possibly inflict using Doomsday's weakness to magic and the ability of a Stand to squeeze internal organs would heal very quickly. Doomsday is also very difficult to pierce hence the ineffectiveness of the Space Ripper Stingy Eyes, and his resistance to cold is even better than that of Superman who lives comfortably at the Arctic Circle.
On the other hand Doomsday would not be immune to Dio's time stop, nor is he smart enough to figure out any way around it. He also lacks abilities like Heat Vision which would be deadly against Dio, and also can't fly. As for speed neither of these two is very fast, and while Doomsday likely does have incredible reaction speed as he can fight Superman, that won't matter when time is stopped.
There is a possibility that The World is faster than light though that isn't really supported by any feat and actually seems to be contradicted by several such as Polnareff's inability to attack the Hanged Man and how Hol Horse's bullet evades Silver Chariot's sword yet Avdol then runs up and pushes Polnareff out of the way, the same Avdol who is later knocked out after being hit by a truck, and the same Silver Chariot that trades blows with Star Platinum almost equally when Star Platinum is the Stand which overcame The World, so generally the Stands being faster than light is pretty questionable but still a possibility, but that also won't matter since The World can't really kill Doomsday and even if The World is slow Doomsday still can't react to it because he can't see it. Stands can only be seen by those with abilities similar to Stands, namely psychic or telekinetic abilities, or some sort of ghostly ability or an ability to communicate with and/or attack ghosts or ghostly creatures, none of which Doomsday possesses.
In the end, Dio will keep freezing Doomsday and landing hit after hit, but he can't really kill Doomsday, but Doomsday will simply get frozen in time before his attacks land, meaning neither is going to tear the other apart. However, once it becomes morning Dio will have to hide, and he'll have to avoid both Doomsday and the sunlight, meaning he'll stop time and run as far as he can to hide in a building, but then Doomsday's going to smash every building looking for Dio, and while Dio can stop time and freeze Doomsday he still can't leave the chosen building because of the sunlight, so once he hides to escape the sunlight it can only end with Doomsday breaking the building and exposing Dio to the sunlight, or with Doomsday failing to find him but that seems doubtful since Doomsday would literally smash the town looking for him.
Note: In canon there is reason to believe that Dio actually no longer possesses his freezing touch, since it's stated by Dio himself that that power worked by controlling the flow of fluids in his own body to use his body a certain way, or more succinctly that he has those abilities because he has total control of his body, and he also states multiple times in Stardust Crusaders that he doesn't have full control over Jonathan's body which is the body he currently wears, since his own was destroyed in the late eighteen-eighties. That would explain why he never uses the freezing touch even though it could be a useful weapon in addition to his Stand. It was factored in here to make him more powerful, plus it makes his moveset more varied and more fun to read.
Also note that while it may seem rude to refer to Dio's pompous monologues as verbal diarrhoea, it is factually true that he actually wastes his stopped time by gloating so that stopping time for eleven seconds means attacking his frozen opponent for one second and spending the rest of that time boasting about his ability to stop time. Him against Doomsday is basically supreme arrogance vs extreme stupidity.
Also note the irony of their names, that Dio wields The World, and Doomsday means the end of the world, so ironically this was the world vs the end of said world if names are taken literally. As a final note Dio actually is seen reading religious writings or at least has been interpreted as such. Granted his reaction to The Bible and the Lord of the Rings and the other books was more the reaction one would expect of Dark Schneider from Bastard, but Dio probably does have that exact attitude to any religious writings, even if he's too regal to voice it. He does consider himself a god.