This post by a Christian blogger well summarizes the thesis of The Search For Christian America, a remarkable book by Mark Noll, Nathan Hatch, and George Marsden, three of the world's most important scholars of religion, who also happen to be traditional Christians. The post reproduces this excerpt from the book:
1) We feel that a careful study of the facts of history shows that early America does not deserve to be considered uniquely, distinctly or even predominately Christian, if we mean by the word “Christian” a state of society reflecting the ideals presented in Scripture. There is no lost golden age to which American Christians may return. In addition, a careful study of history will also show that evangelicals themselves were often partly to blame for the spread of secularism in contemporary American life. . . . (Mark A. Noll, Nathan O. Hatch, George M. Marsden, “The Search For Christian America,” 17)
The post also notes what I've long noted (after of course, those three distinguished authors) about non-authentically Christian ideas riding along side Christian theology:
[T]hey will continue to argue that America does indeed have rich “Christian heritage;” but unfortunately what passed as uniquely Christian, was in fact, Christianity baptized in “Natural Theology,” and rationalist Enlightenment principles. Here is an example of what I am talking about, found in the Declaration of Independence....
The theology of the Declaration of Independence is no more authentically "Christian" than the theological tenets found in for instance, The Book of Mormon. This is something I want evangelicals and Catholics to appreciate. And also as I've noted (again after Noll et al.), John Witherspoon was one of those evangelicals who contributed to the spread of secularism in American life. His Lectures on Moral Philosophy, what he primarily taught his Princeton students like James Madison, did not teach Christian or Calvinist principles, but rather Scottish Enlightenment principles.
Finally you may want to check out this post, another Christian source that thoughtfully explores the Christian Nation idea, where they note something very interesting from one John Eidsmoe, one of the key promoters of the "Christian America" idea:
As John Eidsmoe, one of the more responsible conservative Christian writers on this issue observes: The term ‘Christian’ can be used in two contexts. First, it can describe someone who is “born again,” or “saved,” or “regenerate,” a dedicated follower of Jesus Christ and His teachings… The term ‘Christian’ is also loosely used to denote a person whose beliefs about God, the world, and man are generally in accord with those of the Christian religion but who may not be a dedicated follower of Christ. In this second context, a person’s beliefs, actions, and/or demeanor may be “Christian” (decent, generous, moral) but he or she might not be regenerate. (Eidsmoe, Christianity and the Constitution, 78-79) Eidsmoe recognizes that it is only in a looser sense of the word that the majority of the founders and the origins of the constitution may be considered ‘Christian.’
It seems to me that Eidsmoe lets the cat out of the bag in that passage. Yes America is a "Christian Nation,"...in a nominal, heretical, theologically liberal sense of the term. This is not what the traditional Christians who want to believe in the Christian America myth want to hear.