Is America a Christian nation? A proof

I am teaching a course for elementary education majors on United States history. It is a broad course that deals with the politics of the classroom as well as the content these future teachers will need. We dealt with the idea of religion and the American constitution recently, discussing James Madison's "Memorial and Remonstrance Against Religious Assessments" as a way to defuse the emotionally charged contemporary issue of America as a Christian Nation.

So here is my question: can we write a proof that provides students with a way to deal with the logic of that statement? Here is my attempt.

If America is a Christian nation then it does not possess a civil government. Because if it had a civil government whose existence depended on Christianity then it would no longer be civil.

If this is true then its opposite true, that if American is a Christian nation, then Christianity in America depends on the government's support of it and without such support Christianity would not survive.

But since we know that Christianity would exist without government support, then America's government can neither be a support for it nor exist because of it. Thus American is not a Christian nation.