From what you have written, it does sound like he maybe fancies her a bit. However, that doesn't mean that he will do anything about it. He knows that you are there doesn't he, so his reasoning would be along the lines of 'why would doormat think anything, if she is there with me?' Men can be really insensitive, and take us for granted. However, if you so much as started smiling at another man, I'm sure that he would hate it! Not that I am suggesting you do that of course, as men have a funny habit of thinking themselves innocent of flirting, but hating us doing it!
I would suggest that you don't show that you are cross about his behaviour, but start going to a pub where she won't be! You say he is a friend of hers, but is she yours too? At least if you are in the pub with him, you know that he is not up to anything. Perhaps you could join him when he is talking to her, and be really friendly towards her, so she knows you don't think she is a threat.
Sorry, the last time I was in this situation, I was only 'going out' with dh, and we weren't committed, so, although it was unpleasant, I had no ties to him, or any 'right' to feel jealous.