I've examined this issue extensively on my site's Separation of Church and State page, and can definitely say the U.S. was founded on Christian principles, for the following reasons:
A) Congress has always opened with prayer.
B) All states including the original 13 reference God in their state constitutions.
C) The original version of the Constitution referred to God as the "Great Governour of the Universe", the Articles of Confederation.
C) Most of the original 13 states, 8 of them, kept non-Christians from running for public office.
D) Many of America's universities and colleges began as Christian seminaries for instructing pastors.
E) Separation of Church and State was originally designed to protect the Danbury Baptists who were being imprisoned at the time for their beliefs by Virginia's Anglican state church. Most states were run by churches, as was the case in Europe. Jefferson's letters to the Danbury Baptists are the origination of the phrase wall of separation where separation of church and state comes from. He and Madison united to help defend the Danbury Baptists.
F) Jefferson and Madison both wrote legislation on separation of church and state invoking reference to God and Christianity. Jefferson in the Virginia Statute for Religious Freedom begins by saying:
"Whereas, Almighty God hath created the mind free; That all attempts to influence it by temporal punishments or burthens, or by civil incapacitations tend only to beget habits of hypocrisy and meanness, and therefore are a departure from the plan of the holy author of our religion, who being Lord, both of body and mind yet chose not to propagate it by coercions on either, as was in his Almighty power to do, That the impious presumption of legislators and rulers, civil as well as ecclesiastical, who, being themselves but fallible and uninspired men have assumed dominion over the faith of others, setting up their own opinions and modes of thinking as the only true and infallible, and as such endeavouring to impose them on others, hath established and maintained false religions over the greatest part of the world and through all time"
Madison in his legislation, Memorial and Remonstrance Against Religious Assessments argues for Christian principles as the basis of religious freedom:
"We remonstrate against the said Bill, 1. Because we hold it for a fundamental and undeniable truth, 'that religion or the duty which we owe to our Creator and the manner of discharging it, can be directed only by reason and conviction, not by force or violence.' The Religion then of every man must be left to the conviction and conscience of every man; and it is the right of every man to exercise it as these may dictate. This right is in its nature an unalienable right. It is unalienable, because the opinions of men, depending only on the evidence contemplated by their own minds cannot follow the dictates of other men: It is unalienable also, because what is here a right towards men, is a duty towards the Creator. It is the duty of every man to render to the Creator such homage and such only as he believes to be acceptable to him. This duty is precedent, both in order of time and in degree of obligation, to the claims of Civil Society. Before any man can be considerd as a member of Civil Society, he must be considered as a subject of the Governour of the Universe: And if a member of Civil Society, do it with a saving of his allegiance to the Universal Sovereign. We maintain therefore that in matters of Religion, no man’s right is abridged by the institution of Civil Society and that Religion is wholly exempt from its cognizance. True it is, that no other rule exists, by which any question which may divide a Society, can be ultimately determined, but the will of the majority; but it is also true that the majority may trespass on the rights of the minority...
12. Because the policy of the Bill is adverse to the diffusion of the light of Christianity. The first wish of those who enjoy this precious gift ought to be that it may be imparted to the whole race of mankind. Compare the number of those who have as yet received it with the number still remaining under the dominion of false Religions; and how small is the former! Does the policy of the Bill tend to lessen the disproportion? No; it at once discourages those who are strangers to the light of revelation from coming into the Region of it; and countenances by example the nations who continue in darkness, in shutting out those who might convey it to them. Instead of Levelling as far as possible, every obstacle to the victorious progress of Truth, the Bill with an ignoble and unchristian timidity would circumscribe it with a wall of defence against the encroachments of error."
G) Public officials have been sworn in from America's earliest days on the Bible.
H) The Constitution was based on William Penn's Province of Pennsylvania in 1682 which originally created religious freedom, a bill of rights (then called Charter of Principles), made a two-house legislature which was the basis for the House and Senate, had a governor instead of president, granted trial by jury, had a public education system, and peacefully coexisted with Native Americans unlike other colonies - Penn even insisted on fairly purchasing land from them despite his charter.