America is not a Christian nation. The claim is either trivial, unintelligible, or false.
One might argue that America is a Christian country simply because a plurality of its citizens self-identify as Christians. The religious right is either making a much more substantial claim or committing a logical fallacy.
Theocrats often use the "Christian country" claim as a key premise in arguments of the following form:
(p1) The United States of America is a Christian country.
(p2) Christianity abhors usury.
(p3) Christian countries must not permit anything abhorrent to Christianity.
(C ) Therefore, the USA must not permit usury.
Suppose (p1) means “Christians comprise at least a plurality of American citizens.” On this weak reading, the argument derails, even if we grant (p2) and (p3). So what if a majority of Americans are Christians? What matters is whether a majority of Americans vote to ban usury and whether the proposed anti-usury legislation is constitutional. The weak version of (p1) doesn’t do any work. If that’s all the “Christian nation” claim amounts to, the argument reduces to a civics lesson.
Consider a stronger reading of (p1): “The constitution of the United States of America requires that laws conform to Christian doctrine.” Alternatively, “The constitution forbids any laws that violate Christian doctrine”; or “The constitution requires that we pass all and only those required by Christian doctrine.”
The sample argument makes a certain amount of sense on the strong reading of (p1). The strong reading has a significant drawback, however—namely, that (p1) comes out false. The constitution just doesn’t say anything like that.
Some Christians point to the influence of Christianity on laws, traditions, and secular institutions. These arguments don't have the same force as appeals to constitutional law. We can always ask why we should continue to abide by Christian tradition if the majority now wills otherwise, or if Christian tradition turns out to violate the constitution.
Here’s a medium strength (p1) that Christians often deploy when pressed: “The constitution is based on Christian values.” This claim is too vague to sustain our sample argument.
The values of the constitution are consistent with many of the values of Christianity, but also with the values of many other religions and many secular ethics. The critical point is that the constitution does not appeal to Christian doctrine to justify authority. I.e., the authority of the constitution does not rest upon tenets of faith, revealed truth, or the dogma of any particular religion.
Medium strength (p1) is at best incomplete as it stands. The fact that the Framers were influenced by Christian ideas doesn't imply that they intended to create a Christian nation. If we want to talk about the intellectual heritage of the Framers, we also have to acknowledge their debt to the secularism of the Enlightenment, to deism, to the anti-clericalism of the French Revolution, and so on.
"America is a Christian nation" is an empty slogan. It doesn't mean much and nothing follows from it. It reinforces tribal solidarity among believers and marginalizes non-believers. If you take the claim in the spirit in which it is offered, the clear implication is that non-Christians are bad Americans.
Don Herzog is grappling with similar issues at Left2Right.