The answer is yes. He doesn’t use the term, but he embraces the concept.
In a 2008 debate, Dawkins called human beings “earth’s last best hope,” and acknowledged that we have moved beyond naked Darwinian “short term greed,” to the point that we have the ability to act “on behalf of life as a whole.” From Dawkins’ opening statement in the debate:
Far from being the most selfish, exploitative species, Homo sapiens is the only species that has at least the possibility of rebelling against the otherwise universally selfish Darwinian impulse…If any species in the history of life has the possibility of breaking away from short term selfishness and of long term planning for the distant future, it’s our species. We are earth’s last best hope even if we are simultaneously, the species most capable of destroying life on the planet. But when it comes to taking the long view, we are literally unique. Because the long view is not a view that has ever been taken before in whole history of life. If we don’t plan for the future, no other species will…
In other words, we are exceptional.
No doubt, Dawkins disdains the concomitant imperative that puts us on the pedestal of moral worth because that smacks of religion. Indeed, he has pined for the discovery of a chimp/human hybrid species with which we could interbreed precisely to prove we are not special. But once you admit human exceptionalism, the logical philosophical consequences that flow therefrom become unavoidable–which is why so many across a wide array ideologies from animal rights, to bioethics, to radical environmentalism, work so hard to deny that the differences being human makes are any more morally significant than the elephant’s exceptional trunk.
Bottom line: Dawkins believes human beings are no longer bound by Darwinian imperatives, the consequence of which is to impose moral duties upon us. It seems to me that once our uniquely human duties are accepted, it becomes very hard to deny that we also possess uniquely human rights.