You might need to give your friend's mother some time so she can calm down and not be so angry with both of you.

Is there an adult you know who could help mediate the situation between you and her mother? I mean, someone who is an impartial party (not for or against either of you) who could help you reach a mutual decision about what to do so that everyone feels it's fair? If so, that might be a good place to start.

My other thought is if you agree to see your girlfriend only when there is someone else there - like in a group of friends (or even only in front of her mother, if you have to) - then maybe her mother will eventually say ok to that.

I agree that you're probably going to have to talk to your friend's mom about what happened, but I think doing it in person may be better. You'll have to just suck it up and face her, even if you're really nervous about it. Stand up straight, look her in the eye, don't mumble and let her have her say - and then tell her that you understand why she reacted as she did, but that you give her your word nothing physical happened. Apologize for what you did and let her know you have learned your lesson and ask her what you need to do to prove to her that you can be trusted with a second chance. Good luck!