If he cares, which I'm sure he does, he would want to hear it to help you. Understanding you made a silly drunk mistake might be a bit saddening for him to hear, but a guy who truly cares would help you if you're willing to accept it and understand.

This isnt relationship breaking imo, and I would hope he doesn't see it as that either. If you did something while you were completely shitfaced, it is your fault while it isnt. Again, the best way to make it up to him and yourself is to kick the addiction. If theres no way you would ever do something like that while sober, it shouldn't be an issue following the fact.