I am well aware that both Chomsky and Fodor reject Darwinian evolution.
Is his claim that Noam Chomsky rejects Darwinian evolution true, and if so, on what grounds does Chomsky do so?
Chomsky was famously called out by Dennett in Darwin’s Dangerous Idea, who saw “signs of Chomsky’s agnosticism — or even antagonism — towards Darwinism”, and added that "if Darwin dreaders want a champion who is himself deeply and influentially enmeshed within science, they could do no better than Chomsky".
This is over the top, but some Chomsky's comments can be interpreted as skeptical not of the evolution itself, but of the natural selection as the chief evolutionary mechanism, especially when it comes to the evolution of language: "It is perfectly safe to attribute this development [of innate language structures] to "natural selection", so long as we realize that there is no substance to this assertion, that it amounts to nothing more than a belief that there is some naturalistic explanation for these phenomena." [Language and Mind, 1972] "It surely cannot be assumed that every trait is specifically selected. In the case of such systems as language or wings it is not even easy to imagine a course of selection that might have given rise to them. A rudimentary wing, for example, is not "useful" for motion but is more of an impediment". [Language and Problems of Knowledge, 1988]
Liberman speculates about the underlying reasons:"My own impression is that Chomsky has always been motivated by rationalist epistemology, and has always rejected the idea of Darwinian evolution of mental abilities, which he sees as a sort of genetic empiricism. He simply and consistently dislikes the idea that language might be learned, whether by neurons or by genes. As a result, he prefers to wait for "physical mechanisms, now unknown" at the level of cell biology, or necessarily-emergent properties of complex-enough brains, or some other now-mysterious form of explanation".
The short answer is no, Chomsky doesn't reject evolution; the longer answer goes like this: Chomsky is arguing that it doesn't give sufficient insight into the nature of a 'natural organ of language', the existence of which has been won, as admitted by Dennett in this review of his book by the biologist Maynard Smith, who also agrees with Dennett on this.
Searle points out that Chomsky moved linguistics from a classificatory science to a structural one; that is all human languages have a deep structure that is universal; and this universal grammar is already innate to the mind - in other words the child's mind is not a tabula rasa (blank slate), on which language is inscribed; but already has a language native to it; which when exposed to a language environment - say Gaelic - gradually loses its generality and becomes specific; and it's specific character is Gaelic.
Interestingly, Chomsky ties this to Descartes notion of Innate Ideas derived from Platos notion of *anamnesis in Meno); and is part of his, as Lieberman puts it in @Conifolds answer, rationalist epistemology; this he argues was an inspiration for Kants own epistemology.
One can adopt in fact some of Kants language to make the case clearer: a transcendental argument for something, is for Kant one which derives the necessary conditions of making this something a possibility; here it is the possibility of language, in all it's many-mannered variety, and the condition for this - Chomsky argues - is an Inate Idea of language - his universal grammar - in the human mind.
It's for this reason Chomsky is dismissive of Darwins Theory; he surely accepts its role in evolving this facility; but this in itself doesn't say very much; Chomsky argues that "it is hard to imagine" how such a faculty developed, compared to a birds wing; to which Smith suggests in attempting to make sense of this statement, that
languages don't leave a fossil record whereas wings do
At the end of Chomskys own defence against the specific objections by lodged by Smith; for Chomsky writes:
The frantic efforts to defend 'Darwins Dangerous Idea' from evil forces which regard it neither as dangerous, nor even particularly controversial, at this level of discussion, hardly deserves comment.
ie he understands evolution; but for his own argument it's irrelevant; to put in different, but analogous imagery; one might be hear of a physicist, bothered by why literary theorists hardly ever mention atoms, given that novels are written in books, and books are made of atoms - might suggest literary theorists should take note of this fact; to which literary theorists would retort, quite correctly, that the physicist hadn't understood the point of literary criticism, and the fact that novels are written in books which are made of atoms, though entirely correct, is irrelevant.
That of course no-one hears of such a physicist, is that the irrelevance of this argument to the aims of literary theorists is entirely obvious.
No. He does not. He does, however,reject certain interpretations by certain vanguards of neo-Darwinism of what Evolution is, and how it is to come about. Chief among these vanguards are Richard Dawkins and Daniel Dennett.
And he is hardly the only one to do so. Stephen Jay Gould, one of Chomsky's ardent supporters, was one of the most majorly influential figures in the history of Biology and was opposed to both ultra-adaptationism, cladistics as well various neo-pseudoscientific applications of Social Darwinism, including Sociobiology and Evolutionary Psychology. The latter, essentially a new-Eugenics developed by intellectual charlatan Steven Pinker,is particularly venomous and misinterprets Darwinian principles and applies them to systematically support sexism, racism, white supremacy. This usually stems from a monolithic application of "selection" as the post-hoc justification for certain social imperfections, without taking into account the societal factors that pollute the "system under observation" (things that will be controlled for in a real Scientific enterprise).
Hillary Putnam made certain observations, misguided of course, and was corrected by Chomsky. In Chomsky and His Critics Putnam corrects his mistakes, and pays a fitting tribute to Chomsky's enormous contributions. Similar events occurred with John Maynard Smith. Below, is Chomsky's response to Putnam's blog post which the OP mentioned. I also link it here:
In my entire life, I have never once even hinted at rejection of the modern theory of evolution, Darwinian in origin though of course greatly extended and crucially modified since. You’re quite right in dismissing talk of “innate language,” just as I have always done. I have never once even hinted at your story about the terms of human language, though I have discussed the fact that the modern adherence to the referentialist doctrine, abandoning a long philosophical tradition, has led to great confusions in modern philosophy and is quite inconsistent with basic facts about human language. As for the origin of the terms of human language, as I’ve repeatedly discussed, that remains a deep mystery, for which no one has any sensible account.
Sorry that our efforts to get together have so far failed. Hope we’ll have a chance to talk about these (and many other) things one of these days.