So, he didn't leave against medical advice. He was discharged. That's how anyone not leaving against medical advice leaves the hospital.
If you decide that you don't want care and you don't have a medical need for staying, then you can request to be discharged. A "voluntary" (air quotes) discharge, of sorts. This is simply called "being discharged".
This happened to a colleague recently. Had something stuck in his esophagus that he couldn't get out and couldn't get it to go down. Went to the ED. Puked it up like 20 minutes after being admitted. He asked to leave since he didn't need to be there anymore. He was discharged.
It would include “check himself out”. You either leave against medical advice or you are discharged.