I am open to all comments on improving my blog, but due to hundreds of prior diatribes on the loathsome idea that the USA is a Christian nation at all, I have to say in advance, my following discussion does consider newer ideas of atheistic social contracts in its second section, which won't make sense unless you read the first section.
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The Declaration of Independence, on which grounds the USA is an independent nation, states that the USA rebelled against the British for violating natural rights. Most people believe those natural rights are OBVIOUSLY not based on a belief in God, because Franklin changed Jefferson's original description of natural rights as ‘sacred and undeniable’ to ‘self-evident’ truths. And on the surface intuitive, self-evident truth may appear sufficient.

But intuitions differ as to what natural rights should be, or on how to resolve their conflicts, or on whether rights should exist at all. For example, France holds that its motto 'freedom, fraternity, and equality’ should be self-evident as natural rights too; and many gun owners, wishing to protect their property, feel their liberty should take precedence over another’s life, whereas others don’t. However, the USA has generally been more successful in resolving such conflicts of intuition to the benefit of the nation. because the USA’s natural rights are reasoned deductions from the natural human condition.

Franklin also changed ‘God’ to ‘Creator’ in one place of the Declaration of Independence, to incorporate John Paine’s belief that science makes nature function purely automatically since the Creation.
But in an earlier instance Jefferson’s original words were retained, ‘Laws of Nature and of Nature's God,’ because Jefferson chose the natural rights we now have from John Locke’s expoundment on the State of Nature under God. While some fiercely object to any notion of God at all, it’s not possible to remove it, because:

• The USA declared its independence from the British because the British violated these natural rights. Natural rights cannot be altered without undermining the legitimacy of the USA to govern itself as an independent nation.
• The Supreme Court uses the Founders’ intent to resolve Constitutional disputes. When he was President, Jefferson named Locke as one of the three greatest men who ever lived—Locke’s portrait still hangs in his Monticello museum, preserving his estate for posterity (the other two are Francis Bacon and Isaac Newton). So the Supreme Court must take Jefferson’s choice of Locke’s theory into account when interpreting Constitutional law for the current day.
• Jefferson’s choice is the only reason we have rights at all under a system of positive law (which is law to increase freedom rather than impose restrictions). No other political theory has been able to create a rational system of positive law.
• Jefferson’s choice enables inclusion of ‘pursuit of happiness’ as a right, which no other political theory has been able to do.
• Jefferson’s choice of natural rights was based on well-reasoned deductions from the natural human condition ('empirically') by John Locke, widely regarded as the most influential early enlightenment thinker. This enables resolution of many Constitutional disputes by reasoned deduction, so the Supreme Court cannot ignore Locke’s influence on Jefferson’s choice and consider law to be self-evident instead.

Locke was countering an earlier view of the State of Nature by John Hobbes, who held fear and chaos to be the natural condition, and punishment by a God-fearing, feudalist monarchy as the only solution (“Leviathan,” http://www.gutenberg.org/ebooks/3207). Locke built on Hobbes’ idea of a social contract, whereby individuals trapped in a base state of nature (without authority) imagine an exchange of lost freedom to authority, in order to maintain the social order and escape the natural state of fear. (Many people confuse this with an actual contract, but the social contract can only be imagined, because humans have never existed without some form or authority, tribal or otherwise.) To enable democracy, Locke substituted hope for fear in Hobbes’ earlier version of a social contract, in an extensive argument for the USA’s natural rights, roughly summarized as the following hierarchy:

• Premise: All people are created equal in the Eyes of God, who benignly seeks to grant us justice in heaven. Therefore governments should treat all people equally on Earth.
• Primary: The right to life is necessary so that God may judge people in the best way possible from their own acts in a natural life. Besides declaring killing as the worst possible crime, this enables the State to legislate on water, agriculture, housing, and health.
• Secondary: The right to liberty is not only so that people can find the pleasures they enjoy, but also so that people may choose to act for the greater good, which is necessary for a successful government. Acting for the greater good is rarely rewarded in the current life, so to make these rights a basis for a rational system of justice, we must presume that they are rewarded in the afterlife. This entitles the federal government only to impose restrictive laws that do not restrict liberty.
• Tertiary: The right to pursue happiness is now possible, due to the above right to liberty, enabling taxation for social benefit beyond the scope of the most minimal restrictions on liberty that a state could impose. These include education, libraries, museums, national parks, public radio and TV, etc.

The passage defining the empirical derivation of USA' natural rights is Locke's 150,000-word "Essay Concerning Human Understanding” (https://oll.libertyfund.org/titles/locke-the-works-vol-1-an-essay-concerning-human-understanding-part-1), Book 2, Chapter 21, On Power. The rights are based both on earlier analysis of equality in the same book, and on his earlier and better known work “Second Treatise of Government” which discusses how property could be a natural right (http://www.gutenberg.org/ebooks/7370).

Of course, natural laws are abstract, so Constitutional Law derives legal rights and restrictions from the natural rights, mostly in the Bill of Rights. That is, the authority to rule, granted by the USA's independence by natural rights, promulgates authority to the Constitutional Law defined by the government. When conflicts arise in Constitutional Law, the Federal Supreme Court looks up to natural law for resolution, especially in the Founders’ intent.

Human Rights versus Natural Rights

Since Jefferson, many objections have arisen to Locke's necessary premise of the existence of a benign God. Human Rights attempt to define universal rights that do not require assuming the existence of God. In the 300 years since the enlightenment, human rights has found rational ways to define and prioritize rights to life and liberty, but not to pursuit of happiness. Thus, logically, proponents of human rights wish to privatize schools, libraries, museums, parks, etc. However they have not been successful in convincing the public that their conclusions are better than Locke's, and as things are, the natural rights chosen by Jefferson from the ideas of Locke remain the only rationally complete guide to resolution of Constitutional disputes, which the Supreme Court looks to when considering the Founders’ intent.

Notwithstanding the Supreme Court making decisions when evaluating the Founder’s Intent--etc, etc--there is also the prior existing self-evident interpretation, based on intuition. So when the USA is faced with dissent at a level which threatens National Security, it has sometimes decided to use human rights, rather than Locke’s natural rights, to define decisions. Instead of rationality, human rights inevitably ends up accepting Rousseau’s definition of a social contract as government by consensus, drawn from the notion that human beings are no more than noble savages, who will rebel if a majority has a sufficient complaint. However, human rights cannot provide a complete rational explanation for the Jeffersonian rights, especially with respect to conflicts in rights to life (lethal self defense, abortion, the death penalty) and in how the government should enable pursuit of happiness (schools, museums, libraries, parks, public radio and TV, etc).

The human-rights shortfall is unavoidable, because human rights do not allow religion to define what those rights should be--A problem known as Hume’s Guillotine, first described in his “Treatise on Human Nature,” Book 3, §1 (http://www.gutenberg.org/ebooks/4705) and later by Moore’s naturalistic fallacy (“Principia Ethica”, 1903, http://www.gutenberg.org/files/53430/53430-h/53430-h.htm). Human rights can define rules to provide a safe life, but without any way of being able to say what people should do otherwise, human rights are reduced to the necessities of life itself, such as water, food, shelter, and control (human rights finds government necessary because our natural state, as a species, is a member of the Ape family, and even Apes have authority structures). The logical result of allowing human rights alone to define law is a minimal state, where the government does as little as possible. Historically, however, minimal-state nations often fail, frequently due to revolution. Democracies thus do not consider it wise to adopt a complete minimal-state government, and all current democracies have one or another religion at its basis.

Thus in the USA, federalism allows states to reduce natural rights more than required at the federal level via State Supreme Courts, and individual states decide the amount that atheistic notions of human rights replace Lockean natural rights at the federal level. In the last century, federalist notions on reducing national control on all issues have handed over more and more responsibility to the 50 individual states.

France, incidentally, just accepts Locke’s conclusions. In 1762, Rousseau did write about Locke's views on the "Social Contract"(earlymoderntexts.com/assets/pdfs/rousseau1762.pdf, trans. Bennett, 2017) in which he most notably explores civil reaction to illegitimate power; but Rousseau had nothing to say about natural rights. After the French Revolution and Napoleon, France now feels rights to life and happiness are so justified AND so self evident, they don't even need to be stated!