Only if you're speaking colloquially. We get at least one thing in our own language wrong in almost every sentence we speak and there's no room for that in German.
He is the subject whereas him would be the object. If you rephrase the sentences slightly then you can easily hear which case it should be if you're used to the language. "Who did this?" "He did this"
We can get away with it in English of its SVO (subject verb object) sentence structure meaning we can pretty much use either the subject or object version of the word and as long as it's in the intended place in the sentence, it shouldn't break comprehension. German word order is very fluid though, so unfortunately the correct case muat be used in order to be understood.
It all depends on the "Who?" part. It is incomplete and before we can determine whether "he" or "him" is the correct form, the missing parts have to be filled in.
Ex for "he" "Who [worked on the project]?" "He [worked on the project]."
Ex for "him" "I turned in the report." "Who did you give it to?" "[I gave it to] him."
In everyday American English, to hear "he" or "she" without a verb of some kind is unusual to the point of nonexistent. "Him" and "her" are much more likely to be used without any other words.
Most english speakers would expect to hear "him" in a situation like this because, while the male being refered to is the "subject of discussion", he is not necessarily the "subject of the sentence [fragment]".
Grammatically, he is more likely to be the object of the sentence, which is fragmented here, but a way to help understand would be to flesh out these phrases:
"Who [are you referring to]?" "[I am referring to] him!"
"Who [are you in love with]? - [do not tell me that it is] him! [my brother!]"
It's not clear if there are supposed to be two speakers or one person is answering their own question as a way of seeking confirmation or clarification. Either way, in English you would most likely want to use "him" not "he".
You can use "he" (as subject) to imply something like this, "Who [is your doctor]? He [is my doctor]!" But that is a lot of specific information to leave unsaid or try to imply. So usually, in English, when someone just says, "Who?" They are likely implying, "Who are YOU referring to?" in response to something you have just said.
Who (was it)? (It was) him! Who (did it)? He (did it)! Both are grammatically correct, depending on which verb is implied. However, I would always use "Him!" and never "He!" as a full sentence, because "He!" just sounds wrong to me.
Another example is "Who (are you talking about)?" "(I am talking about) him!"