"Yes, he does."
Translation:Ja, det gör han.
The reason you can't say "ja han gör" is that gör requires an object. So if you compare to 'make' instead it might get clearer – you wouldn't say "Yes, he makes" in English.
gör is not quite like 'does' in the English translation – 'does' is an auxiliary verb, but gör is a normal verb that can be used as a replacement verb instead of repeating the main verb. In English, you could say 'Yes, he does sing', but in Swedish, you cannot use gör together with another verb. Also of course you use 'do' in English to create questions, like 'Does he sing?', but gör is not used like that in Swedish.
"Ja, det gör han" is the way we Swedes say "Yes, he does". " Ja, han gör" doesn't really mean anything in Swedish, it's not a full sentence. In this case, the question could be "Äter han?" meaning "Is he eating?" and the answer then "Ja, det gör han" with "det" pointing at "äter" = "eats", as in "Yes, (that) he does". "Det" could mean both "it" and "that", depending on context.
'Yes, he does' is the answer to a question like 'Does he …?' For instance: 'Does he speak Swedish? - Yes, he does'. In Swedish, we say this as Talar han svenska? - Ja, det gör han. This is the standard, idiomatic short answer to questions of this kind.
Ja, han gör det should normally be translated as 'Yes, he does it/that'. But it can also work if you want to say 'Yes, he does' (implying perhaps that someone else does not). It can actually have a third use too, if you stress gör a little bit, it sort of means 'Yes he does do that'. So I've added it as an accepted answer, but don't use it as the standard answer to short questions, because it will sound broken.