I don't think there's any easy answers here. You just have to figure out what you can/can't live with, and stick to that.
I don't know if this would be worth asking him, but you might say, "You told her that you wanted to leave me. Do you really want to leave me? Why do you want to leave me?" Perhaps there are deeper problems in the relationship between you and him that he hasn't talked about.
It's not okay for him to be dishonest -- to you or her. If you decide to stay with him, it should probably be on the condition that he's going to tell (both of you) the truth from now on.
I don't personally think "veto power" is evil per se, but I do think it should be used with utmost caution, and only under the most strenuous circumstances. However, the your circumstances here are pretty strenuous, so I don't blame you for considering this veto. Just make sure you know this is a hard boundary for you before you go through with it. There is always the notion of, your husband screwed up, can he be trusted to do better in the future?
Sorry things have gotten this bad.
Love means never having to say, "Put down that meat cleaver!"