Your thread title says it all... if you really like your gf and want to be with her then try to make it work... apologize and learn from your mistake..
Act like it didn't happen and nothing is wrong. Because, with women, the moment you start apologizing they immediately think the opposite of what you intended was your true feelings. Words are words, but women pick up on the non-verbal cues. But beyond that, based on how nervous you are about this situation, I doubt your girlfriend cares about it, one way or the other, because she already knows that she has you under her thumb, but she will make you think that she is mad just to get you further under her control.
Here's a clue: girls only care about cheating with guys that they are subordinate to, not the other way around. Why would she care about what you are doing if she knows already she has your balls.
judging by the problem you described and your writing, im guessing you both are in high school? if so, i would suggest not having a serious relationship at that point in your life anyway. i understand the need for guaranteed hook ups, but the variety of hookups you will have over the course of your high school and college careers is even more valuable. i made the mistake of having a serious relationship through those years and have regretted it ever since. now that im working its hard to meet new girls and they are always looking for something serious. take advantage of your age and freedom. think of this as a blessing in disguise.