No. Fallacy of the false dichotomy. There is literally a whole universe out there that does not have to be explained by either "God did it" or "man did it."
Fundamental rights are simply a necessary condition of being human. How rights, or any other philosophical construct, can exist a priori without resorting to a God who created them is a fairly complicated area of philosophy I won't delve into; suffice it to say that the simplest explanation, "God did it," is not the only coherent and logically consistent one. Mathematics flows from a few basic postulates, without regard to whether those came from an omnipotent creator or not.
"We hold these truths to be self-evident" -- not revealed by scripture. "their Creator" and, elsewhere in the Declaration, "the laws of nature and of nature's God." These are references to natural law, not to a particular religious tradition or a particular conception of the Divine.
God is the same yesterday, today and tomorrow. He will never change and he will never take back the Rights he gives us.
"to preserve these rights, governments are instituted among men, deriving their just powers from the consent of the governed. ...[W]henever any form of government becomes destructive of these ends, it is the right of the people to alter or to abolish it, and to institute new government, laying its foundation on such principles and organizing its powers in such form, as to them shall seem most likely to effect their safety and happiness."
This construction holds true whether the laws of nature exist as a synthetic a priori or as the gift of a benevolent God. If they are laws of an impersonal Nature, there is no universal assent to what those rights are, and the task falls to the people to determine what they are and how best to preserve them. If they are ordained by God, there is no universal assent, and the end result is the same.
Thought provoking post. Very well written.