Though Dudley Dursley wouldn't admit it, he had been profoundly affected by the Dementor. For the first time in his life, he had encountered something that tantrums, threats, or fists could not conquer. No, the Dementors invaded his mind, feasting on hidden insecurities and rampaging through his memories as though gleefully devouring a particularly violent picture book.
Also for the first time, he realized that perhaps his cousin was not a scrawny, scar-faced waste of space. He began to experience a rather peculiar emotion, one that was, he would realize in his adult life, gratitude. It was that same foreign emotion that prompted him, on the day of their planned-unplanned-replanned departure from his home of roughly 17 years, to leave a cup of tea outside Harry's door.
Contrary to his mother's vehement belief, Dudley was never a very intelligent boy. Perhaps, if he had spent more time exercising his mind and less his muscles, he might have been a trifle less like the blundering idiot Harry's lot presumed him to be. But why, he reasoned with his limited capability, waste time with your head stuck in a book when you can lift weights and knock out the wanker who thought reading was a better idea?
Anyway, his restricted mental capacity (or perhaps something unknown) failed to connect his departure from home to his departure from Harry. It was not until Harry stood facing him, standing beside that queer lot of Errors, or whatever (rather than beside Dudley himself), that the young Mr. Dursley realized life with his cousin had ended. It was simply too ridiculous a concept to process. Harry had been a gnat, buzzing irritatingly round his head, but fun to smash, for as long as he could remember and now…Now he was off to battle some Voldy-something-or-other. They must be daft—seventeen-year-olds don't fight mass murderers!
Still feeling that the whole thing was some sort of dream (and, honestly, not knowing quite what possessed him), he thrust out his hand for Harry to shake, and then that was that. He'd never been eloquent—a nice crack to the jaw was so much more effective than mere words—and even if he had been, he wouldn't have been able to summon more than the simple declarations he made that day. What, after all, did one say to the cousin he hated that had nonetheless saved his life?
In his later years, Dudley thought back now and then to that day, and his fingers hovered poised over some generic holiday card. His insides squirmed just the tiniest bit, but he chalked it up to indigestion and resolved never again to order whatever particular brand of take-out he'd just consumed. In the end, he always wrote simply "Merry Christmas, Dudley." And then, as such moments of reminiscence always made him want to pummel something, he'd head to the gym for an hour or two and take out his frustrations on the punching bag.