Universal human rights result when a deontology is taken as a prescription.
(Later edit: I realized I did not clearly weigh in on the actual question. The answer is ambiguous: you can demonstrate that proposed rights meet the axiomatics of the deontologies involved, and that most people from most cultures would naively sign on to those in some simplified form, the same way we readily accept naive set theory.
I can demonstrate slavery, or child labor, is always unfair the same way I can demonstrate that the continuum is uncountable. But not in the same way I can demonstrate an electric field produces a magnetic force.
History or reflection might give you a different way of looking at infinity. It might also leave you with no reason to trust any of my instruments. So ultimately, neither kind of demonstration is really superior to the other.)
For a Kantian/Rawlsian or other ethics that recognizes 'duty' and assumes it must have a rational basis, if something can in fact be implemented in such a way that it honors all individuals' autonomy properly, it is a moral law, and it might just as well be an enforced law. A lot of Enlightenment thought comes down to some echo of this kind of thinker, often via the notion of a social contract and the rights of people to enter, alter, or leave it.
If everyone honest is going to ultimately agree that a rule treats them properly, then you might as well cut to the chase and force the issue. Whatever you have a duty to do or not to do creates the right of all others to have you do or not do it.
We all recognize the idea that one should not kill innocent people, whenever possible. It is not something that is culturally determined or foreign to anyone. So as an international point of agreement, we can declare it to be a guiding principle. Starting from a few examples of that kind, you can discern the pattern Kant supposedly reached by abstract critique, and start making agreements that things that meet certain standards are natural to humans.
But, as is kind of notorious within 'duty' arguments, almost all meaningful duties are contingent -- they contain variables that individuals get to choose according to their best perception of their own autonomy. Is it necessary to sacrifice children to gods?
That depends upon a cultural history and what role the sacrifice plays in keeping that culture in existence.
So virtually nobody can enforce any of these principles in a way that is really helpful. All they can do is use the underlying agreement to criticize one another's contingent choices. This is still useful in extending a certain kind of peace, but it privileges those who are already advantaged. Diffuse rules generally serve those privileged to be in a position to withstand considerable waste.
For instance, there are those who can legitimately afford peace and those who may actually perish unless they take what others have. Making the assumption that the notion of peace has a very high value suggests that those latter cultures should really just not exist. You can obligate the privileged to help them, but it is highly likely that they don't ultimately want outside help. They want the chance that nature gave them, whether or not the only chance they naturally have is moral in others' eyes.
So the idea kind of eats its own tail. Individual obligations may be impossible to protect from group needs to define themselves and sustain their identity, and vice versus.