Should I feel bad about leaving a woman at a bar alone with the check on a first date?
That's really rude. Why did you do that?
I was starting to dislike her from the moment she told the waitress she didn't drink alcohol.
You shouldn't abandon people on dates.  When you invited her out, you were implying that you were going to pick up the check.  Your date has her own reasons for not drinking alcohol, and you shouldn't have instantly hated her for that.
So I should stay and finish the date.
Absolutely, you should stay! If you don't have a good time, then be honest with her about not wanting a second date.