My best friend's son is a Navy nuke. He has apparently attempted suicide, and is being processed for that, but is home on leave. My friend and his son had a massive fight due to the son's bad behavior while home, which is when the suicide attempt was revealed, and the son is apparently "losing his job."

I spent ~3 years working around nuclear weapons and nuclear power, and the stress from working in that environment day in and day out takes a toll. We had a lot of people in my command get processed for drugs, alcohol and stress related issues. I 100% get it. I personally got to the point where I absolutely didn't care about shit, and did drugs and drank way too damn much. I was lucky enough not to get caught. I never considered suicide.

My primary concern is how the Navy handles this situation so he gets the help he needs. Does anyone know how this works? I'm going to be hanging around with my friend and his son for a couple days.

He'd He's your best friend's son. How close are you and the son? Would he want you knowing he attempted suicide? I wouldn't say a word about it unless it was brought up first by them.

I hung out with him today, and we did quite a bit of shooting of his guns. We talked about a bunch of stuff. He's a little over halfway thru his enlistment, and definitely not planning to reenlist. 14 more years of aircraft carriers and instructor duty does not appeal to him. I confided in him that I had some pretty dark days and that I'd done some dumb things when I was in, but got through it by talking with friends. I let him know that he could talk to me if he ever needed a friendly ear.

You are a braver man than I to go shooting with someone who had recently attempted suicide and was still in the process of their life falling apart.  Especially someone who was still exhibiting behavioral issues.

Depending on the circumstances, his support after discharge could range from absolutely nothing to significant VA support.  Nobody would be able to give you a reasonable guess without more information; likely information that he would be unwilling to share (or may not be honest about).