Do you have to break up? That is a silly question. Of course you do not HAVE to break up. There is no law or rule that says you must break up when your partner does x, y or z. The question you really mean to ask is should you break up.
And honestly, this whole things seems like a really silly reason to break up with someone. So he flirted with someone that he had a bit of history with? Big deal. Apparently neither of them took it seriously. Had she sent him a picture then maybe, but they both seemed to take it as he says it was intended. He seems to have accepted that he made a mistake and that he offended you, so why the drastic response of asking to break up?
Good golly, if you are going to break it off over something like this, then you might as well become a nun because no one is going to live up to the standard that you seem to be requiring.