The only just foundation for government is the Natural Law. To the extent that any government is just is in proportion to its adherence with the Natural Law.
Until 1776, governments were usually formed by kings who pacified rival tribes and principalities by superior strength. Then, at some point, the nobility or the people assert their authority and slowly impose the principles of Natural Law. The United States of America is exceptional because it was explicitly founded on the Natural Law. The founders first used the principles of the Natural Law to justify our separation itself from Great Britain and then used them to constitute its government. The leader, George Washington, was then elected according to the Constitution, and more exceptionally, stepped down voluntarily.
There is much debate about whether the USA was founded as a Christian nation. The Natural Law is the western, Judeo-Christian name for what C.S. Lewis describes as the Tao. Unfortunately, while the Natural Law is written on the hearts of all, it is only in the West where the Natural Law was recognized, described, debated, and most importantly, validated by Christianity and the Judaism. In that sense, as we are Natural Rights republic and Natural Rights are intrinsic to Christianity, yes we are a Christian nation.
In a larger sense, though, what would a Christian nation look like? The first commandment of Christianity is to love God, but how does a government enforce this and still be Christian? Love is a commitment (not a feeling) stemming from an act of the will. The nature of our conscience, then, is one of liberty. That is, the nature of our conscience is to seek God which also means that the nature of our consistence is to get lost sometimes, even deliberately. To be just, a government must respect the intrinsic liberty of our conscience. At the same time, the greatest common denominator for governing humans is the Natural Law and governments are obligated to lead us towards it.