I recently heard a sentence: "Wow, can you not?" A friend stated that it was missing a verb. I said that Can worked as the verb in that sentence, and then he responded that Can could be used as a Main Verb but in this usage it was an Auxiliary. So, is "Can you not" an elliptical construction or a complete sentence?
If the other speaker in the conversation expresses an inability to do something, it is unnecessary to repeat what it was. Can you not? is enough. Yes, this is an example of ellipsis, in which elements of an utterance can be recovered from some other part of the discourse. That doesn’t mean that can becomes a main verb, though. It can never be anything other than a modal auxiliary verb.
In the construction "can you not," can is actually be a modal adverb. Modal adverbs are used to describe the state of truth of a verb. "Would," "Could" and "Will" are other examples of modal adverbs.
In this case, the implied sentence is "Can you not [do something]?" Because the verb 'do' is implied, but not actually present, it is an elliptical construction.
The term 'sentence' is not well-defined, in that there are different definitions (a 'minor sentence' is not a sentence by most definitions, for instance).
What you say your friend says (it's 'missing a verb') can mean at least three things:
(1) 'It's a statement that may be considered to be formed (by ellipsis of the verb) from "Wow, can you not jet-ski?" say.'
(2) 'It's a statement using a construction sometimes known as a sentence fragment – it's not a traditionally-defined sentence, as it lacks a main verb.'
(3) 'You shouldn't use a statement using a construction that's not a traditionally-defined sentence – a main verb is always required for grammatical acceptability.'
(3), which is quite possibly what is implied here, is wrong.