Let's say that science proves that God exists

[Esquilax]

Only in a philosophical sense. If the universe was intentionally created for our existence then we are more important than the rest of the universe that was created for our existence. If on the other hand we are the accidental unintended by product of mindless forces we can hardly say we have any rights on that basis. It's not binding on anyone anymore than the notion we should treat others as we would be treated is binding in any sense. The USA for example is a secular government tied to a theistic philosophy as noted in this phrase from the Declaration of Independence.

We hold these truths to be self-evident, that all men are created equal, that they are endowed by their Creator with certain unalienable Rights, that among these are Life, Liberty and the pursuit of Happiness. That to secure these rights, Governments are instituted among Men, deriving their just powers from the consent of the governed

The reason its important in this instance is because these basic rights are not granted by the government or the state, we are endowed with these rights by the Creator. The role of governments is to secure those rights. If the rights were granted by the government then they could be taken away by the government.

Is that seriously the extent of your argument here? Please tell me you have more and I've just missed it.

Because leaving aside the fact that just because the words show up in the constitution doesn't mean there's actually a creator for rights to come from, just that the language was formulated to pander to a mostly theistic society, there are more countries than America. And they all have doctrines that aren't the Constitution. Some of them even contain language that doesn't mention god or any creator. Are we to then conclude that those rights are given by the government, but they're given by a creator in any society that mentions a god in its founding document?

You're arguing that rights are some objective thing, but when you provide proof you fall back on an inherently subjective measure. It's a little shaky.