This is the concept that there is an objective moral code established by a Creator and grounded in human decency. A natural law by which both man and state should be judged and which neither is above.
No one before the Enlightenment would've recognized this principle in precisely the way you're formulating it, though
any polity claiming divine legitimation for its laws regulating human interactions--a state of affairs which long preceded Constantine--could be said to fit the broad outlines of the description. By way of illustration, and to return to my earlier comparison, Aquinas didn't share Locke's views concerning the right--indeed, for Locke, the duty--of citizens to overthrow a government not operating by "natural laws" (and again, this concept of "natural laws" goes back to the Stoics). Because, for Aquinas, rights aren't an inherent property of individuals, but rather a necessary legal consequence of the community's collective pursuit of Virtue (similar to Aristotle's view).
This is why I was trying to get you to specify something beyond "the Judeo-Christian philosophy" as a justification for the granting of rights; that's a uselessly broad and vague label which clarifies almost nothing about which ethical model(s) a person is in fact applying. After all, the preference utilitarianism of Peter Singer and the modified Kantian deontology of Tom Regan (both prominent animal rights advocates) are themselves ethical models from within the "Judeo-Christian"--really, Western--tradition.
Look, I'm not really expecting you to be able to pin your case against animal rights on one consistent philosophical model. I don't support animal rights myself (though I'd support some expansions of cruelty-prevention laws) and I can't claim philosophical consistency in opposing them; how to rationally justify extending rights to all humans while categorically excluding nonhuman animals is famously among the most intractable problems in modern Western philosophy. (To return one last time to Aquinas, the only earlier Western philosopher I know of who directly addressed the question, "it is not wrong for man to make use of [animals], by killing or in any other way whatever [since God created them expressly for human use]...Charity does not extend to irrational creatures." By which we can safely infer he'd have opposed even cruelty-prevention laws, 'Charity' otherwise being his justification for obligations towards the rationally incapacitated.)
But, if you're going to carry on about "the Judeo-Christian philosophy" and how any argument for animal rights is inevitably fated to "torpedo" it with devastating human consequences, then in that case you'd better be able to spell out exactly which ideological strand of "Judeo-Christian tradition" you mean to stake your claim on, and demonstrate perfect consistency in applying it against these 'torpedoers'. Otherwise your case for "human exceptionalism" basically boils down to, well, Discovery Institute talking points, which aren't known for their philosophical rigor.
Right, because all humans must be seen as equal regardless of physical or mental capabilities and all of equal moral worth.
If we've learned anything it's that anything less opens the door to all sorts of horrors.
...For instance, assuming this is your answer for 'Why grant rights to mentally impaired humans?' , it sounds like negative utilitarianism in the vein of Karl Popper, not the modified Lockean view you seemed to be hinting you favored earlier.
Socially, Western Civilization's moral code and theory of human rights developed under Constantine in the 300's A.D. and progressed through Thomas Aquinas, The Magna Carta, Blackstone's Commentaries and the U.S. Declaration of Independence.
I've never heard of a history of ethics or rights in the Western tradition that takes Constantine as a starting point; he didn't revolutionize existing Roman thinking or law on the rights of the various categories of subjects. Are you singling him out because as a Christian he granted full political equality to Christians? He also introduced new restrictions on Jews--conversion to Judaism was made illegal, and the price of converting a Christian to Judaism, in particular, was being burnt alive; Jews could not marry non-Jews; Jews, but not Christians or pagans, were forbidden to own slaves; etc.--so he couldn't be credited with introducing equality before the law for all religions, rather with laying the groundwork for the political privileging of Christianity within the Empire. (It was left to his successors to introduce the first laws limiting the rights of pagans, as the needed political capital to do so wasn't quite there in Constantine's time.)