It's possible to get the medical wristband for overdosing, but it doesn't tie into the story or his character as well.
Those are the bandages they give you if you attempt suicide by slitting your wrists. If you survive the attempt and the doctors are able to save you, you'll definitely have that bracelet.
That's why he didn't want to raise his sleeves up earlier in the movie, that's why Andy reacts more softly to that revelation: "You need help Gary"
Andy knows Gary is an alcoholic, Andy knows he does or used to do drugs. That isn't a surprise and doesn't tie into the themes of the film or the conversation they have. The emotional climax here is when they reveal the reason for Gary's fixation on the Golden Mile, and the depths of his depression that led him to attempt the ultimate act of escapism from his misery.
This and the scene just before it in The Hole in the Wall are two heartbreaking scenes that give such incredible depth to his character, and create more empathy between me and a fictional person than I can recall feeling.
This is art