I think it's a mistake to look for an exact moment. It probably happened fairly gradually, though I think there was a strong attraction from the moment they met. After all, the Doctor made it a point to come back after telling her to run away just to find out her name and introduce himself.
The Doctor had obviously fallen for Rose by the time of "World War Three." That scene in the Cabinet Room says it all: "I could save the world but lose you." And the Dalek in "Dalek" obviously knew that the Doctor had feelings for Rose by the time of that episode. I think Rose probably had feelings for the Doctor by then and wanted him to open up to her about them and pursue her, which was why she brought Adam aboard the TARDIS -- to spark some jealously in the Doctor.
Rose had definitely fallen for the Doctor by the time of "Father's Day," IMO. They were way too angry at each other for both of them not to have fallen for the other by that point. In particular, Rose's line -- "For once, you're not the most important man in my life!" -- sticks out.
Rose and the Doctor were definitely dancing around the elephant in the living room that was their mutual feelings for each other by the time of "The Doctor Dances;" Rose seems to have felt that the Doctor, even if he loved her, was either incapable of or unwilling to engage in a romantic relationship, though she seems to have been trying to get the Doctor to change there. Each was jealous of the other's other attractions by the time of "The Parting of the Ways."
Mind you, neither one was willing to admit to their feelings, and the regeneration, I think, made things weirder for Rose -- even as it also made her more physically attracted to the Doctor than she had been already.
ETA:
If you really want to single out one moment, I suppose I'd point to that moment in "The Unquiet Dead" when the Doctor turns to Rose, grabs her hand, and says, "I'm so glad I met you."