The Bible does not force anything but the moral law on unbelievers.
It works the same way ancient Israel was commanded to treat “strangers”.
Our laws against murder, theft, etc., are based on the Old Testament moral law. Pre-American European law - the idea of what’s right and what’s wrong - was based on the Bible. This is why sodomy laws were called sodomy laws, because that’s the word the King James Bible uses. The founders did NOT discuss what the moral law of America would be, i.e., should we make a crime and what we should not make a crime from a secular standpoint. In that day in Europe and America the Bible was the generally accepted root source for understanding what is right and wrong.
The guys who wrote the Constitution corporately came up with this open-ended idea on religion which only worked until the selection of elected leaders started to go into the toilet, which was very early on.
1st Amendment is too vague and allows Christianity to be attacked. So it is being attacked.
Post-WWII we started to pick and choose to legalize some acts, such as sodomy, and murder of unborn children.
Just because a majority wants to make something legal, that does not nullify the effect of the moral law given in God’s Law Word the Bible.
There is no middle ground on religion even though secular humanists maintain that there is. This is being demonstrated now as the secular humanist government turns on Christianity - and Jews who are practicing - in favor of what self-serving politicians see as an up-and-coming group, islam. A nation is either Christian or it’s not. Jewish or it’s not. Muslim or it’s not. Secular humanist or it’s not. If it doesn’t say what it bases it’s law on, it’s secular humanist; it bases it on what the power struggles between people result in. Dems in charge, abortion is legal. Repubs in charge, abortion frowned upon. Believers in charge, abortion illegal.
This is why, IMHO, the real battle is for hearts and minds of the population.
IMHO, it’s the truth that sets us free, and lying to ourselves that sins of the flesh and other sins are acceptable does not set us free, it makes us a slave to those sins.
The Bible tells us that civil government has “the power of the sword” to restrain evil. If it doesn’t use it righteously evil is not restrained.
While I am sympathetic to your statement it needs some explanation and nuance. To simply create a category called moral law and to separate it from other laws of God is to infer that the other laws are immoral or amoral and is to accuse God of being capricious.
God requires His subjects, that is His creation to obey His laws. He has divided human society into institutions that He created and ordained. Those institutions are the family, the civil magistrate and the church. None of the institutions are subordinate to the other, they have interaction and responsibilities, and some limited intersection with the others in their respective service of God, but none is to be master of the other, they all report directly to God. God has given His law in written form (the Bible) as a moral imperative and as a societal blueprint.
Some of the laws are in the form of thou shalt not and honor your father and mother, but much of God's law is given in case law. Sometimes principles are given from which applications are derived and sometimes applications are given from which the principles must be discovered. If an honest exegete believes a set of laws to be applications of a greater principle then it may be honestly stated that while the principle stands the applications change because the circumstances change.
Two examples to illustrate. Deut 22:8 requires that a barrier be put around the edge of a roof so that no one is likely to fall off the roof and be killed or injured. We no longer entertain guests on flat roofs as was customary in the culture of the Old Testament so the law would seem to be no longer applicable; however the law is an application of a greater principle to take prudent measures to protect life and limb thus the principle stands but applications are different. Perhaps we might lawfully (according to the principle of Deut 22:8) require a fence around a swimming pool or an open trench. Another law (Lev 12:2)requiring that a women who is to be treated as unclean for seven days after the birth of a male child and if a female two weeks is a much more difficult question. It appears to many as a law that we might place under the heading of separation laws, that is customs for the covenant people of God to demonstrate their separation from others and into grace. New Testament thinkers are conflicted on these laws as to their current status and application. I personally believe that in the New Testament even dietary laws which were also of the separation type have changed for the new reality of the risen Christ and the blessing to all nations, other separation laws have likewise been changed to reflect the redemption accomplished in Christ.
There can be no conflict in the mind of a Christian that God's laws are law and that His judgements will stand even when they are in conflict with USSC. In other words the first commandment trumps the first amendment. BUT it is not entirely clear if the civil magistrate is to enforce the first commandment and if so what remedy is has been given. Psalm 2 instructs civic leaders to fear God and obey His laws, Jehoshaphat was commended for removing idols and purging homosexuals, Gideon broke the alter of Baal and was made a judge, Jeroboam was condemned for idolatry that he made Israel to sin, and many other examples demonstrate that the general welfare is served when the nation is lead by men who fear God and that has not turned away form the worship of the true and living God and ruler of creation.
Yet, the law of God not only institutes the civil magistrate it also limits it and here is where theologians of all stripes disagree. Back to the matter of public nudity. I am unaware of a Biblical law given to the magistrates either by principle or application that outlaws public nudity. But the general welfare of a people is best served when they do cover up. This can be well reasoned and exhaustively demonstrated from natural law even if revealed law provides no specific application. So even if this were a "Christan" nation I do not think that automatically makes public nudity a moral crime, but neither does it forbid the use of natural law to order society for the general welfare as long as the jurisdictional limits given by God are respected.
As for the Founders, they did in fact expound on the limits of government.
A wise and frugal Government, which shall restrain men from injuring one another, which shall leave them otherwise free to regulate their own pursuits of industry and improvement, and shall not take from the mouth of labor the bread it has earned. This is the sum of good government, and this is necessary to close the circlue of our felicities. - Thomas Jefferson.
So no... You cannot incorporate Leviticus into the USC. Sorry.