I know Korean grammar usually requires sentences to end in a 'verby' word like an action verb (동사), descriptive verb (형용사), or 되다 (to become) or 이다(to be something). But can correct, full Korean sentences ever end in something else?
It is reasonable to say that since full, grammatically correct English sentences (besides the usual one word "Yes.", "No.", and "Hello." expressions) always come with some form of verb, the same can be said for Korean.
However, comparisons to English might not be so sound since Korean and English are quite different. So, let's jump into something more universal: linguistics!
The notion that a sentence needs to have a verb or adjective is slightly incorrect only because several types of sentences exist. Thus, it's only fair to say that in any language, grammatically correct sentences can come in verb-less forms (See "Major and minor sentences" section of the wiki page).
But, in terms of simple sentences (sentences that do contain at least one subject and one predicate), it would be against that very definition of a grammatically correct sentence to come in a predicate-less form.
TL;DR Yes and no
Korean sentences can end with a verb 한나는 집에 왔어요.
verb copula 저는 학생입니다.
or an adjective 유나는 정말 예쁘다.
Further information can be found here http://www.linguajunkie.com/korean-2/make-korean-sentences