Yes. Utterly. And proof of its correctness is that it generalizes. I am a composer. I originally only composed in a classical style (using the same basic tonality you find in Mozart, Beethoven, etc.). But then I tried my hand at other genres, including jazz and even atonal percussion music. And I found that I could make the transition, i.e. I could transpose my ability to 'speak classical' into these other musical idioms, the reason being that, despite the differences between them, the same basic structures were being instantiated.
Be it noted that, in this context, when I refer to Chomsky's 'Universal Grammar Hypothesis', I am referring not so much to his views about prepositions and clause barriers and the like, all of which is prohibitively boring, but to the underlying conception of mind; specifically, to his contention that different languages are instantiations of a single biologically rooted, innate grammar and also to generalizations of this principle, such as the just mentioned one concerning music.
The fact that Chomsky's theory holds with respect to music tells us something about music. Exactly what it tells us is not immediately clear (as in, I would have to spend a couple of hours figuring it out, though not much more than that), and that something is something that hasn't yet been delineated.