To answer the OP's question, it is important to recognize that legally, the U.S.A. was not established as a Christian nation. It was instead based on innovative, seventeenth-century ideals of Deism and freethought. The founding documents of the United States, namely the Declaration of Independence, the Articles of Confederation (no longer in effect) and the U.S. Constitution do not invoke Jesus Christ (or His Father, for trinitarians keeping score) as the reigning deity, or reference Christianity as the civic religion of the newly independent republic. And this was not implicit, either; those documents were written in an age when the English Parliamenters were required to give their oath of office "on the true faith of a Christian", and explicit declarations of faith in political documents were common, and expected.
Instead, we see that the Declaration invokes a generic, unnamed Creator as the author of human rights and the inspiration of our moral insights. It tells us also that this Creator gives us leave to violently overthrow any civil authority that is not established by "the consent of the governed"; a notion that certainly is not promulgated in the Sermon on the Mount or the Epistles of Paul. These Revolutionaries audaciously invoked Divine Command Theory to justify a revolution against a divine-rights monarchy, at a time when the orthodoxy of established religious sects would have no toleration for such a notion. They did this, not as a clever rhetorical trick, but because they earnestly believed in Deism--in a monotheistic Creator deity who reveals himself exclusively through the built-in reasoning faculties and moral conscience of everyday individuals, and not through the miracles and preachings of larger-than-life messiahs, priests, prophets, and kings.
We can see more of this Deistic approach in the writings of Paine and Jefferson. Paine wrote his Age of Reason after the Revolution. Jefferson snipped out all of the miraculous out of the New Testament Gospels and pasted the remainder into a literary piece he called The Life and Morals of Jesus of Nazareth; an endeavor he described as recovering diamonds from a dunghill. And Paine was even less forgiving in his descriptions of the Christian faith.
The New Testament gives very little practical advice on handling political power or administering a state ("my kingdom is not of this world", etc.). Thus, Christian rulers have borrowed extensively from the Old Testament ethos, perhaps out of political expedience rather than theological wisdom. Perhaps it was inevitable that popular sovereignty and freedom of expression would not exist until a secular, Enlightenment-era regime took over. It is certainly established history that countries under Catholic rule stifled religious and scientific insights, but those under Protestant rule fared little better. John Calvin, for example, ordered a theologian executed for publishing a treatise On the Errors of the Trinity.
On the other hand, one could argue (and many have argued) that the relative success of the American project is due to its Christian population (and the same has been said about many other prosperous countries as well). This is not inconsistent with Scripture, and could well be true. One could also argue that an environment where there is no temporal power associated with religious institutions encourages true piety, and is therefore more favored by G-d. A reading the 1786 Virginia Statute for Religious Freedom gives a sympathetic view to this argument.
Christianity and democracy both are subjects one could study individually for an entire lifetime. It's a complicated question, and one that can't really be answered without a lot of opinion and guesswork.