Trust issues are going to be something he needs to work on himself. That's his issue to own. You can't make anybody trust you, and unfortunately his self preservation instinct may be making this harder. Regardless, it's not fair to you to be constantly accused of infidelity. That is going to lead to resentment.
It is OK to set a boundary with him regarding his accusations and irrational mistrust. You do not have to tolerate it if you have never done anything to betray him. If he continues to do this, you don't have to stick around in the relationship to be his target.
Also, I can understand the issues about having a social life as a supporter. Your sufferer may not want to socialize at all, or even leave the house. It is unfair for him to not want you to socialize without him either. You have to have a little time to do the things you enjoy too. He needs to understand this and be sensitive to your needs as well.
With that being said, if he is a priority for you, you can probably kiss most of your social life goodbye. He may not capable of handling noise, crowds, and people like you. If you want to spend the majority of your time with him, a lot of it will probably be alone and in a place he feels safe.
Just an aside... Arguing with a vet with combat PTSD is like throwing gasoline on a fire. They were trained to "fight" in fight or flight situations, so arguing does no good. It's better to not engage, and wait until they are calmer to have a discussion.
