She’s a slip of a girl, hardly more than a child, and she’s human, which makes things even worse. Humans have such short lives compared to Time Lords. The Doctor knows it’s foolish, but Rose is so alive, so vibrant, full of curiosity, courage, and wonder, that despite his better judgement, he’s smitten.
It can never work between them; a time will come when he’ll have to leave her behind, just like everyone else he’s travelled with, but not just yet, and hopefully not for years.
For now, what does it hurt to let a little romance into his hearts?