Linguistic innatism has been a fairly confused notion, in the sense that there can really be no question as to whether or not language is innate. It's a one-step conclusion licensed by the observation that every human on Earth, barring pathology, acquires a language. The real contention is over whether or not whatever allows humans to acquire language in the first place is specific to language (domain-specific) or general to other cognitive capacities and mechanisms (domain general).

Other animals have domain-specific capacities that no one really contests, and humans have non-linguistic domain-specific capacities as well. In this sense, the question really boils down to an empirical matter.

Chomskyan generativists take the empirical facts surrounding (a) what children come to acquire given input in their environment, (b) the structure of that input, and (c) the mechanisms that underlie grammatical processes, properly understood, as all pointing towards domain specificity.