I don't think you were wrong for shying away from the birthday party because of problems with that other girl. You didn't want to put yourself in a bad situation, that is VERY mature of you. And, even though you did something contrary to what you wanted to do, I think the fact that you went to the party anyway because of your friend's feelings says a lot of mature and positive things about you as well.
At the age you and your friend are at, it's very possible that it's time for you to take different paths. There's no sense in you trying to keep up a relationship with someone who doesn't want to show you the same respect and care. I don't think that you should confront her and say "we're not friends", but just do your own thing, for you. Pick up a new hobby, join a club or a team at school, make new friends! People drift in and out of your life through out it, and some stay out, that's just how it is. Just live for you and maybe someday she'll be one of the people that drifts back in, but don't fret over it much either way.
Best of luck.