There really is nothing much you can do apart from what you are doing already. Your instincts are sound, now you just need the patience of a saint to get the job done.

He knows he can trust you. That's why he wants you there. It is not really neediness or worry that you'll leave. You make him feel better by your presence.
If he didn't you'd be turned aside with a shrug and a grunted "I'm alright." The same stuff he pulls when you say he should go to the doctor. Basically, the doors to manliness city slamming shut in your face with a deafening boom.

Getting him to talk though ... you realise you are trying to get a guy to talk about how he feels, without the benefit of alcohol, truth drugs or raging hormones? Most guys get uncomfortable with that at the best of times, never mind when there is something actually wrong. Stupid, I know, but it is there to be dealt with.
You are going to have to take your time and walk a very narrow path. Don't pry, but let him know you are listening to everything he says. Most therapists agree that the silences are often more eloquent than the words used.
Let him know you want to help, but don't seem pushy about it. That is guaranteed to back him into a corner and peeve him.
Don't let get him drunk, no matter how tempting. Drink and depression mix about as well as gasoline and lit matches.
If he hasn't already, try to get him to see the doctor about it. That is not going to be fun for you at all, but the sooner it is treated, if necessary, the faster recovery will be.

And tell him, from me, an ex-depression sufferer, to man up, grow a pair, stop being such a wuss and admit there is some sort of problem here, then get it fixed.