I agree that its worth having a fight over. If I were in a relationship and the other person did this, I'd be upset, want to make sure they are really over ex, talk about our boundaries with others, figure out if we are on the same page, etc. (I know people who've had sex with their ex's. I would be concerned).
What makes this red flagish to me is, new man says he can’t trust me and he doesn’t know if it’s fixable. So rather than taking the incident as a bad judgement call, he sees it as a character flaw. May be it's just a word choice, but the words used seem unreasonable for what happened.
Some emotionally abusive men love love love to have something that they can use against a woman to keep her in her place, to keep her under control, to keep her walking on eggshells, to limit her contact with others. This situation would totally work for that. So my advice is sure, own the behavior wasn't wise / showed a lack of appropriate boundaries / whatever, but nothing more. If he can't get over it, then good riddance.
Even asking how to earn back trust is an odd question -- she's asking how she can turn summersaults for new man to convince him she's not shagging everyone else.