I confronted him when he came upstairs. He claimed that nothing happened between the two of them. He was just talking trash and trying to be macho. He told me he only said those things to his cousin to "save face." He said he did not want to look like a punk for not sleeping with the woman so he put on this facade. He had no desire or intention to sleep with the woman. He was drunk.
Your H is behaving like a typical wayward blaming all his bad behavior on someone else. The fact that he talks this way as a married man to others should be very alarming to you. He wants to present himself to the world as a single man rather than a married man.
I would have a huge problem with that because it is profoundly disrespectful to you.
Being "honest" about being bad does not make it GOOD. If I murder someone and am "honest" about my crime, it does not erase the damage done. "I was honest!" is the refrain of someone who wants to use their honesty to erase their crime.
My suggestion would be to have a discussion about this attitude of his and how alarming and hurtful it is to you. I would also point out that if his cousin finds this kind of talk COOL, then she is an enemy to your marriage. There is nothing MACHO about a married man acting single, it is wormy, unmanly behavior.
Secondly, I would affair proof your marriage. That means not going out without each other. you are married, act like it. And most importantly, I would be snooping like a blood hound to make sure there is never any contact with this woman again. You would want to make sure there are not any others.