Yes of course your relationship can survive, but you need to understand thoroughly why it happened, and deal with her immense insecurity to make sure that it doesn't happen again. No-one is perfect, very good, earnest people sometimes do very horrible things which they wish with all their heart that they had not done and would do anything to take back.
If this is the situation with your girlfriend then if you both work at it, without reprisals or judegments, together you can of course come through the other side of this, sometimes far stronger.
Don't allow pride and hurt destroy something this precious to you, at the end of the day this is in the past now and after relevant discussion about how to help her gain self esteem and about maybe making sure she doesn't go out and get hammered so that her judgement is impaired, you should leave it in the past.
I guarantee you that in ten years time this won't matter at all if the relationship is good from heronin, but it will take commitment and work by both of you to ensure that happens.
Don't let the person you love slip away because of a mistake.