realistically i think neither of you will lose your job and this will carry on for months.
i also think if you leave it it won't get better.
i also think it's something personal between you and him, so you should speak to him about it and tell him that you guys are going to be together in this work place for a long time, and it would be better for both of you if you understood each other better.
tell him you know you make mistakes, tell him you know you can be difficult, but you're doing the best you can to be the most pleasant and hard working person you can be. tell him you're doing this because you DON'T like running to the boss, that you don't like holding a grudge and that you don't like having people angry/disappointed with you.
you can't ignore it, unless you win the others over to your side and alienate him