Men just aren't as articulate as women. That doesn't mean we don't boil with emotion. But sometimes, verbally sparing with a woman is like putting together IKEA furniture while wearing mittens. It's frustrating.
Sometimes I think relationship fights should be settled by skee ball.
That said, I would return the favor when he goes silent. He wants control of the situation, and he's getting that. He's not resolving the argument, or communicating, or getting to the core of why the two of you are disagreeing. He's doing only what it seems he can do -- control the fight by shutting down. It drives you crazy, right? That's the point. He probably also hates the drama of it all and will do anything to just stop the madness.
Here's the thing about relationship fights -- everybody loses. Both the man and the woman. Because it's the opposite of communication. It's a sign of two people being uncompromising, refusing to listen, grow, and forgive. And love is forgiveness... within reason.
So I recommend returning the silent treatment and waiting him out. Drop the topic. Give it some time. Let him think and stew. Give him some space. I'm pretty sure your campaign to get through to him has short-circuited his BROgramming (Gawd, I'm a hack.)
And then, one day, whilst snuggling on the couch, get to the point. Don't be accusing or mean or emotional. Just straight-up tell him "I was really hurt/upset/angry the other day because of X. I'm not now. But I want you to know why I was that way. Maybe next time, we both won't end up in a squabble." For the record, I'm not suggesting you hide your feelings and let them turn into tumors. Just... back off.
I'm assuming, of course, that the reasons you two are fighting are misdemeanors. Those little wrinkles that you think are gaping wounds.