Yes, although there are other theories. Some of those other theories actually retain some of Chomsky's ideas, such as our innate capability to learn language (which seems to be the idea most commonly disagreed with). Two examples are Lexical Functional Grammar (LFG) and Head-driven Phrase Structure Grammar (HPSG).
Why would that idea be disagreed with? Even if language learning started out piggybacking on some general mechanism, that general mechanism would have biases, and evolution would, I think, tend to optimize/specialize it over time.
I can't really speak for why it is disagreed with--I personally find the claim that language is innate completely convincing. But one reason I've heard for disagreeing is the notion that some animals have language (not true unless you water down the definition of "language"). Others seem to think that general learning mechanisms explain the acquisition of language. I find that outrageously naive, since virtually all children acquire language (and if they're raised bilingually, more than one language), whereas even good linguists struggle to define all the rules of a language.
