I'm looking for advice from either people who have been in my shoes or in on the other side as I haven't encountered this before. Sorry if this gets long....

I work for a big public company and manage a big team. I have 3 direct reports and 14 indirect reports. One of my direct reports has been with the company for a little over a year now. He started off pretty well but about halfway through his first year, his performance went way down. He also manages a team of his own and I've seen his team struggle. I've had many talks with him as we were previously meeting at least once a month and now are meeting on a weekly basis. The issues I had with him stemmed from him rarely meeting a deadline but also not communicating that he was going to hit said deadlines. Most of these deadlines have been self imposed. I don't believe he has a really tough work load. However, I've had to clean up a lot of his messes. He's getting a reputation around the company as someone who is not responsive so people come directly to me instead. This is, in turn, holding me back as I find myself having to do his job as well as my own. My boss is aware of this and we're trying to manage.

This all blew up a week or so ago in one of our weekly meetings where he blew up at me for talking about one of his items that was due to a few VP's but he missed his deadlines. He blew up at me saying that he feels like he can't do anything right and that his health is more important and walked out of the meeting. He grabbed his stuff and went home. I was shocked. I know none of you know me personally but I swear I was not mean nor unfair. I genuinely wanted to have a constructive conversation so that we can figure out next steps. Neither of us were in the office the next 2 days as that's when we had the polar vortex so the office was closed. He did send me an email apologizing. When we next talked, I assured him that he doesn't have a target on his back and that if he had health issues than he needs to handle that as that's most important. I asked him to think about what he needs. We have his review discussion coming up today so I'm sure it will be a little tense.

So here's the thing. He came up in FB as someone I might know so I clicked on to his profile. He doesn't have it listed as private and linked a bunch of articles on anxiety and depression. His most recent one was about what people wish work knew about depression.anxiety. I really want to be helpful to him. I think he's a good guy with potential but he's not getting his job done and I don't know how much longer we can be ok with that. So my question to the group is how do I best deal with this going forward?

tl;dr version: I have a team member who is not doing well at their job but I suspect it's due to depression/anxiety. What is my "obligation" as a manager to help him get his job done?