This issue isn’t yours to work on now. You’ve done your part, explained yourself, and your boyfriend is looking for ways to blame you. Scrolling back six months to find a text that said you didn’t want to be rude, but talking won’t change things to your ex isn’t cause for breaking up — unless you want it to be. Your boyfriend found something he wanted to find. Rather than see it for what it was, a statement of disengagement from your ex, he took it as a sign of betrayal. His insecurity, jealousy, and vulnerability has the best of him.
I’d be clear that he will either have to trust you or not, but that the issue is his to solve — not yours. If he can deal with it, that is great. If he can’t, don’t waste more of your time trying to prove yourself. He will find something to think he is being betrayed even when it isn’t the case.
If you want to offer going to couples therapy, I think that would be a very open gesture, but this may be something your boyfriend might want to try tackling on his own. The find help tab at the top of the page will help you find someone in your area.