Within the standard evolutionary worldview, where humans are just part of a continuum of life stretching back to lifeless chemicals in a primal ooze, there is no logical basis for inherent human rights. Any such rights can only at best be a pragmatic social construct to be discarded on a whim.
At any rate, it sounds like you are unaware of the history of your culture.
This is likely a category error. Natural facts don't necessarily have anything to do with moral facts. There are plenty of non-naturalistic arguments for natural rights.
There at least has to be something ontologically different between humans and other creatures for humans to have special status. But everything in naturalism is just reducible to physical laws, so everything is ontologically equivalent. Can't have a moral order with monism, since you need multiple things to have an order.
I don't see why the acceptance of morals facts requires attributing some special ontological status to humans. You would have to be assuming something about animals or humans that doesn't seem entailed by anything discussed thus far.
Furthermore, mathematical monism absolutely is compatible with a moral order and the world we experience, as but one counterexample.
In our finite world of tradeoffs, rights mean one thing must give way to another. For example, plants have lesser (no?) rights than humans, hence we eat plants and not humans.
Without a value ordering, which requires something other than monism, such common sense tradeoffs make no sense.
At any rate, it sounds like you are unaware of the history of your culture.