Dear Nicboc~
You are in a very worrying situation and it is hard for you to know what to do - particularly as your options are somewhat limited.
Having anxiety and depression and an eating disorder that results in pain are all pretty big things and need professional treatment. just by itself the eating disorder, as Kanga says, can be very dangerous indeed.
Any sensible person will understand if someone cares about them and is worried. It's true that depression can make a person want to be alone or not talk, however the need for proper medical support remains. Even if he does get angry it may be worth it if you end up getting him to go to the doctor.
I would suggest two things. Firstly try to get your friend to have medical help, and if he rejects your arguments then see if there is anyone else who might have more success in persuading him. Does he have a family or friends you can talk with about his situation?
The second thing is simply to let him know you are there for him. He may not wish to talk about how he feels, that's OK. Try to do things with him if you can that you both enjoy.
At such a stressful and worrying time do you have someone to care for you? A parent or friend perhaps. In these situations you need support too to talk things over and share the load.
Croix