It could fly as a statement when you don't want to be precise about what has happened to him.
But I think that as a question your examples would work better as "Он ранен?" (in case of violence) or "Он травмирован?" (in case of something self-inflicted and more)
Edit I mean that "Он болен" could cover your both examples if you want to be evasive
I should add that "травмировать" does not exclude intention. "Он специально травмировал нашего лучшего игрока / Он специально нанёс травму нашему лучшего игроку" is absolutely possible in sport too. But "ранил/нанёс ранение" sounds even worse to me. And even this can be an accident. One doesn't know that a gun is loaded and accidentally shoots someone in his leg for example.
It's a bit difficult to define in one comment.
I meant something that was not intended by anyone the way it can be in case of violence (it is not a very precise definition of course). An accident.
I wonder whether my definition of "болен" comes from "быть на больничном". It's when one is officially not fit for work for some time.
@Zeitschleifer: interesting - when you mentioned violence I did not assume that it was deliberately inflicted. Falling off a cliff is a violent accident even if you simply tripped and fell. It indicates sudden and traumatic injury - the body getting smashed, bashed, and broken - rather than presence or absence of hostile intent.
I would not recognize falling off a cliff as a violent accident in English. Something to learn again...
Violence = насилие in Russian. Насилие, насильственные действия in Russian require an object and a subject in my opinion. I don't see it in case of someone unintentionally slipping off a cliff on his own