We seem to disagree on the definition of natural rights. For a right to be natural you require that it will be granted by God or that it will be based on human nature, and by human nature you mean to say that it is a direct product of the evolution biology that have created our species. While I would grant you that it's a definition that you can find in many philosophers, following Moore, but in the current context I think that the natural/artificial distinction isn't useful for defining natural rights.

I would argue that this definition is very narrow and limiting, it introduce weird dependency on our current scientific knowledge, and isn't very useful. For instance, when Hobbes proposed the social contract theory he was discussing natural rights but today we know that his natural science knowledge was incorrect and therefore he was actually describing artificial rights. To me this makes no sense. Instead, I will propose, that rights that are derived by reason, that are universal, and that do not depend on a specific state law or the social norms of a specific society are natural rights. They are natural in the sense that they are not dependent on any state or law but are inherent. Those rights are not granted by god, and they are not artificial law propositions. They are based on universal principals of reason and the reality of human existence.

This view and this definition of natural rights is not my invention. It's reflected in the language of the universal declaration of human rights - which recognizes a set of universal rights. The declaration isn't a legal document that legislate a binding law. It recognize rights that are not (let's hope, are not yet) generally accepted by all nations. Nevertheless those rights are not based on God or born by the act of composing and publishing the declaration, those are natural rights. They are natural despite being in opposition to humans natural behavior, despite their consistent violation. It is because those rights are natural that they can serve as basis and justification for international law and justice.

Rawls theory of rights is universal, it isn't about specific social norms, it discuss human society in principle. One might say that his ethics are based on theory of the human nature.