and another........

If you don't go back, she'll know you were just being nice. If do DO go back, she'll know you don't know how to take no for an answer, and get creeped out by you.

You asked her out, she said no, end of story. Going back to apologize is NOT going to make her feel better. It will make her feel worse.

But then again, what the hell do I know. I'm only a woman.