I. America's Christian identity is cultural, not Constitutional.
Alexis De Tocqueville identified this reality when he visited the young American nation,
"The Americans combine the notions of Christianity and of liberty so intimately in their minds, that it is impossible to make them conceive the one without the other; and with them this conviction does not spring from that barren traditionary faith which seems to vegetate in the soul rather than to live."
Democracy in America, 1835
The Founders were not of one mind theologically. A number were not orthodox Christians and some were accused of atheism and impiety in their day. They did not write any kind of Christian creed into the Constitution. But pointing that out, or cherry picking pithy quotes from Thomas Paine, does not overturn the observed reality that Christian churches were central to community life (including education), that moral consensus rested upon Christian assumptions, that Biblical quotes permeated public discourse and that most of the Founders were Christians.
This broad Christian consensus, Tocqueville noted, allowed for a robust politics because it tempered rather than inflamed sectarian claims. America was spared European style religious warfare not by denying Christianity, but by applying its virtues broadly and voluntarily in community life.
The genius of this approach comes into law in the First Amendment to the Constitution:
Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof...
No religion shall be foisted upon the people through government, nor shall government dabble in curbing the religion of the people. You can't be required to affirm the Apostles' Creed to run for dog catcher; likewise you can't be prevented from gathering some friends to pray for the election of a Christian dog catcher.
America is more diverse today, and the idea of current Christian cultural consensus is debatable. But one has to ignore and falsely revise history to deny the formative influence of Christianity on this nation.
II. Secular Government
Turkey's 20th century revolution established a secular government - its Constitution banned overtly religious parties. Yet Turkey is an indisputably Muslim nation.
Secular government prevents the imposition of theocratic government - but secular government is not "atheistic" government. A government that states "there is no God" is way over into the realm of religion. Secular government does not assume a secular world view or impose one on the people.
Secular government, practiced correctly, operates from the principles of the First Amendment. It does not endorse a religious creed but it does not presume to suppress any creed, either, beyond denying any creed's claims to rule over citizens who don't confess it.
Christians must make common cause with others to influence law. It is not enough to say, "Here's the Christian teaching, so this is what the law must say." Rather, Christians can voice our perspectives and concerns in the discussion of an issue, knowing that action will require buy-in from others who might not share our beliefs but who do share some common concern. Roman Catholic social teaching frequently appeals across differences to "all people of good will."
An example is playing out in New York right now. New York is one of the few states without "no fault divorce," and is considering its adoption. Christians and Feminists, who disagree on many things, are voicing opposition to the change because both groups see it as harmful to the common good.
Secular government, rightly understood, does not impose Christianity as the only voice, nor does it prevent it from being one of the voices.
III. Right now, government over-reach is a much greater threat than "theocracy"
The idea of a theocracy is abhorrent to Americans - it was, in fact, abhorrent to the Founders and the Christian culture that established the United States.
The Founders were aware of the raw power of majorities and built into the Constitution features that would frustrate absolute majority rule - even rule by the Christian culture consensus of which they were a part.
Right now, we have people who call Christians "wingnuts" asserting the near paranoid position that electing someone like Sarah Palin would lead to theocracy. It's a straw man argument, assuming that any cultural appeal to "Christian values" translates into an attack on the Constitution.
The Founders were also aware of the pretensions of elites - minorities with an agenda - to exploit the majority. They wanted this frustrated as well.
But the reality around the country today is that factions are getting over, with government help. While I lived in California, "zoning laws" were being used to favor business interests over churches in some communities. Around the country, people are invoking "harassment" laws to force coworkers to keep Bibles or other signs of Christian identity out of sight. This warping of "freedom of religion" into "I expect government to keep me free from even knowing that religion is around" has entered public discourse. Ironically, it subverts secular government by entangling the state in religious matters. The state is called in to protect the hapless citizen from the religious boogey man.
One form of this warping is the call to protect "cultural" expressions of religion against free expression of religion - with the result that the state actually serves one faith group against another:
America ceases to be a Christian nation when enough people are convinced to be something other than Christians in their community life. The state has no business preventing or advancing that. Those who use the state to attack Christians degrade protections available to all citizens, actually opening up the likelihood of more intrusive and narrow government in the future.