That's an interesting question. I can think of an explanation, but there might be others.
The first is that in the uncontracted version, you'll notice that when you say It is, is bears the stress: in a more emphatic form the vowel in is will be lengthened, and the pitch of your voice will modulate from high to low. Those cues are used by your listener to understand that you are stressing is to place contrastive focus on the truth value of some proposition that's been called into question (sometimes called verum focus). I can't think of any other context where you'd say It is.
However, when you contract it is to it's, the whole point of the contraction is to convert a two syllable iamb into one atonic syllable, so that you can speak faster, or to avoid a stress clash, but it makes no sense to do so when you need to preserve the usual stress pattern of it is.
Notice that you can say It's not (also used for verum focus) just fine, because the stress is supposed to belong to not anyways.