This is not sustainable.
As flattering as it is to be a primary confidant and therapist he needs a NETWORK of people to help him out. The fact that he doesn't want to go therapy and get the help he needs is just as bad, in my mind, as a cancer patient who doesn't go to the doctor because he/she doesn't want to talk about the disease. Frankly, it's unfair to you and to everybody else he relies on for emotional support.
Honestly, his behavior reminded me of my own sister, who either 1) lashes out or 2) shuts down whenever she is faced with a situation steeped in emotion. It's one thing to step away for a moment, it's another thing to duck it entirely and run away.
Both my sister and I were physically abused when we were younger - and although I had to go through major depression with bouts of suicide ideation, in a way I'm glad that I actually went through that and emerged stronger on the other side. It's also good do know that when my depression does rear its ugly head, I have a toolkit that would probably take up an entire hardware store. However, she denied that the abuse ever happened in the first place. Her children no longer live with her, and she has a history of using pot/alcohol/boyfriends to get by.
If he was truly a romantic partner for you, he would try to be the best partner he could be. You would be meeting his friends and the people back home would know about you. Those are two red flags right there that you can't ignore. Why is he treating you as someone he should be ashamed about? Why would you want a partner who would want to keep you a secret?
Disease or no disease, he's not ready to be committed to this relationship.