When faced with an ordeal dealing with love, you should think. Say, you do something wrong, the one you like/love/both-but-you're-in-denial will suffer, and you will not be able to take it. Faced with the ordeal, you must act quickly, but before you act, you must think.
Whenever Kagome screams InuYasha's name—most likely during danger—InuYasha acts quickly. He turns to his instinct, which tells him, 'Save the girl. She's a shard-detector, after all.' But it's much more than that. When he thinks more about it, his instinct tells him that Kagome is not only a shard-detector but a live human being capable of love and care, who also detects shards. And when you slap that on, that's more than enough reason to save her.
However when Kagome breathlessly thanks him, InuYasha's heart swells up. He cannot explain why, but her smile reminds him of Kikyo. Hell, her whole face reminds him of Kikyo. Their personalities are too different however. While Kikyo is more content and quiet about her feelings other than anger, Kagome chooses to act them out. While Kikyo can stay still during a fight, Kagome is moving all over.
But when a being is hurt, the two girls can't help but help the being, no matter who they are. This pisses InuYasha off, and he's always voices how he feels about the situation. The girls' remarks always remind him that the two are different. Kikyo nods. Kagome snaps. Kikyo lives in his world. Kagome lives in hers. They are different, but they feel the same to InuYasha.
That is why InuYasha is now questioning himself. While Naraku is doing some demon hair thing, controlling Kikyo, capturing Kagome, what should he do? Kill Kikyo and save Kagome, or kill Kagome and help Kikyo? While Miroku and Sango and Myoga (who just chose now to pop up) and Shippo scream at him to react, he takes awhile to think. He has no chance attacking Naraku due to his injuries, and his heart is torn in two for the two girls.
So who should he choose?