Can the question why be a one word sentence?

"Why?" is not a sentence, because It does not have either a subject or verb. It is acceptable only for informal usage.

The general rule in English is that a sentence may imply either a subject or a verb, but not both. As others have suggested, "Go!" can qualify as a sentence. The subject (you, the listener) is implied, but the verb, "go," is stated. So it can be a sentence. Another sentence of the same type is, "You!" which means, "You are the one." In this example the subject, "you," is stated, so the sentence only has to imply the verb, "are."

With "Why?" however, both the subject and the verb are implied. When written by itself, "why?" usually means either, "Why should I do that?" or "Why do you think that?" Those have subjects and verbs (I and do, or you and think). Since both have to be implied here, "Why?" by itself, is acceptable only in speech and in informal writing.