That would need another verb, poder = to be able to. "El no puede oir." = He is not able to hear/he cannot hear. There are things I can do that I do not do. ;)
But in Spanish if you can't hear what someone is saying you can say "¡No oigo!" which literally means "I don't hear!" but no one would ever say that. You'd say "I can't hear!" I think "The man can't hear" is a valid translation.
You might use the sentence in this dialogue: -Que paso? -La problema es el hombre no oye! -Porque? Tal vez él no puede! -Sí.
How would you say that the man did not hear, like in this instance, as opposed to he does not hear?
Two translations for this sentence is: The man does not hear. OR The man does not listen. But what if I wanted to say a specific one? Like if I was trying to say "The man does not hear" someone else might interpret as "The man does not listen."
You would have to say the guy over there then the sentence. From what I have learned, Spanish uses a lot of things like this, so you have to specify the context a lot.