Having once been the source of greatest emotional intimacy for him can totally make you feel responsible for giving support now. Understandable. In this case, it's even more important since your ex isn't very social.
But it's not your responsibility anymore, not even as a friend. You are not in the ideal position, either for your own sake or for his, to play that role, because:
1) you might have your own subconscious selfish agenda
2) HE might think you have an agenda, which would just insult/upset him
3) he might accept for the wrong reasons, like just holding onto you longer
Etc. Think of it as a conflict of interest. Wanting to help is commendable, but I don't think you should follow through.
You will probably feel guilty for a while for "abandoning" him, catching yourself wondering how he's doing and wanting to reach out, etc. That's normal. But don't give in. Your ex is an adult, loner or not, and is responsible for himself.
When you're interacting, don't run away if you're supposed to be in the same space, but stay away. Give him a nod, *maybe* with a very small, very brief smile to indicate you think of him positively, but don't hold eye contact for too long. Try not to stare at him, etc. He might interpret that as you still being in love, etc.
Since he'll see you a lot, he'll have the opportunity to approach you and ask for a friendship if he decides that's what he wants. That's kind of a whole other can o' worms though.
I've had fewer relationships than you can count on one hand (can't imagine why...). The break-up I can talk about ended on similarly civil "not working" terms. Interacting with him NEVER helped. I was struck with a morbid curiosity to know what he was doing, etc., but it never made me feel better.