The very notion of democracy is based on Christian principles — a historical fact, though one not really emphasized in our public school system. Yes, the Greeks had a democracy, but it was not a democracy for women and slaves.
The ideal of democracy is such an important aspect of American history, government, and politics that it has become part of how Americans identify themselves. Many Americans also construct their identity around Christian beliefs, so it is understandable that democracy and Christianity would become intertwined and some would insist that democracy itself is a product of Christianity. Some Christians even argue that without Christianity and Christian principles, there can be no democracy.
The above myth expresses a belief that is completely at odds with reality, but it's also something which more than a few Christians seem to take for granted. How can it be a "historical fact" that the "very notion of democracy" is based on Christianity or Christian principles when democracy existed long before Christianity? It is at least acknowledged here (though not always) that democracy was originally a Greek idea, but admitting that democracy was originally Greek is an admission that the first and primary statement is false: democracy is not based on Christian principles.
Christians faced with this undeniable contradiction between their wishes and historical reality try to get around this by arguing that the original Greek democracy was somehow not a "real" democracy like Christian democracy. This frequently requires distorting the historical record, though — here, for example, it is suggested that Greek democracy wasn't real democracy because women and slaves couldn't vote. That's true, but what's ignored and suppressed is the fact that America originally excluded women and slaves from our democracy. For a long time in America, women and slaves not only had no right to vote, but they had no social, familial, or political power whatsoever.
Indeed, what Christians who repeat the above myth fail to realize is that America wasn't even founded as a democracy, strictly speaking, but as a democratic republic. Only members of the House of Representatives were elected directly — every other post in the government was insulated from the people. The authors of the Constitution were very worried about "mob rule" and thus created a government in which democracy played a significant role, but which could also correct for any excesses of democracy. To be fair, though, ignorance of this fact is common in America and is not limited to just conservative, evangelical Christians.
A further problem for the claim that democracy is based on Christian principles is the undeniable fact that, for most of Christian history, Christian nations were not only not democratic, but strenuously resisted democratic change. The concept of "divine right of kings" was explicitly based on Christian principles, but how democratic is that? How many centuries of Christian rule passed before any Christians happened to notice the allegedly unmistakable basis for democracy in their own religion?
Many important Christian leaders throughout history have not only not advocated democracy, but have in fact been very authoritarian. Did Luther or Calvin ever promote democracy? How many popes lived and died before the Vatican agreed that democracy might be a good thing after all?
Where is democracy mentioned in the Bible or indeed in any early Christian texts? Where did Jesus or Paul argue for democracy? How long was it before any Christians decided that democracy might be a good idea? If Christianity is the foundation of our democracy, why isn't Christianity — or indeed religion generally — referenced as such in the American Constitution? If democracy is necessarily based on Christian principles, why did early advocates of democracy have to argue so hard, and sometimes fight against, Christians?
There is a good reason why the idea that "democracy is based on Christian principles" is not emphasized in America's public school system: it's a completely false claim that has no basis in historical reality whatsoever. To teach that democracy is based on Christianity would be like teaching roses are part of the animal kingdom, or that Paris is a city in China. No reliable text or school would ever think to teach such a thing and so misinform students.