Two things I guess. I think someone who has dealt with depression can be successful in medical school. The key thing is though to go in with your eyes wide open, knowing the risks, and have a plan and support system in place. It seems like everyone on here who's done this and been successful has been monitored closely and been proactive about preventing problems or addressing them quickly. Those that haven't been successful seem to be scared to seek help and try to tough it out or struggle through it on their own until the negative consequences (poor grades, total breakdown, etc) force them to face up to it. Studies have shown that there is a much higher rate of depression and suicide in physicians that the general population. The risk is there.
He should have a mental health provider to touch base with throughout the process. If your son's depression is well managed and he's discussed it with his psychiatrist and the psychiatrist doesn't have objections then it might ok for him to give it a go. A good psychiatrist is in a unique position to be able to asses this as they have been through the training process he wants to go through and they have insight into your sons condition.
Second. We'd all like to think that medical professionals would be sympathetic to those with mental health issues. Unfortunately there still seems to be a lot of stigma in the medical field regarding this, despite the high percentage of medical professionals that deal with these issues. Discussions about this and related issues come up in the psychiatry residency forum and other places on here quite a bit. Some will be supportive, others won't. There will still be people, students and attendings, that act like someone shouldn't be in the field with those issues. I still think it's best not to disclose this info to anyone who doesn't need to know during his training, outside of a few very close friends he knows well and can trust.
I'm not sure how possible it is to avoid having people see his scars. At some school's students practice certain things on each other, taking a pulse, checking blood pressure, drawing blood that might require him to expose his arm. He doesn't need to explain how he got them though. Maybe just saying something like "bad experience, rather not talk about it" or "long story" if anyone asks would be sufficient? Like it was already said, it might not be as obvious as you think it is.