I tend to agree with Marc Gravell's answer on this, but I would like to add that I don't think that any religious influence or moral framework could make a true democracy work. True democracy, history has shown, always degenerates into what Lord Acton described as "the tyranny of the majority" in which the rights of the minority get trampled by the opinion of the majority.
I'd also like to point out that the quote that you used was not intended by the founding fathers of the United States to support a democracy. It was used to state that they felt that the government of Britain had trampled certain rights.
The idea of unalienable rights was an important part of providing the form of government that was eventually established here - not a democracy, but a Constitutional Republic with some democratic elements: A form of government in which the Constitution is the rule of law, and the protection of freedoms established in the Constitution were designed, specifically, to limit what the government is able to do, and thereby protect the country from the pitfalls of a true Democracy.
As Marc pointed out, The U.S. is now fractured and divided among many lines, which is, in my opinion, a result of the fact that we've lost sight of the fact that we're not a democracy.
People on all sides seem to have forgotten the fact that we're not a democracy, that might (whether in strength, or in numbers) doesn't make right, and that there's a reason we are supposed to have a limited government. The fact that we've forgotten the idea of unalienable rights is allowing us to slip into the "tyranny of the Majority" and it's only the fact that both sides have approximately equal support that we're not there already.
Also agreeing with other's sentiments, Christianity itself is split among too many lines. This site is a small representation of Christianity as a whole, but even among the Christians here we're split. Even on this site, there is a concern about the tyranny of the majority, and I think it's a valid concern.
So, in summary, I believe the answer is "no". Christianity might be able to provide some good moral foundations, and certain teachings that coincide with what's best in us, but no religion can form as the basis of a government. I personally believe that the idea of unalienable rights, endowed by our Creator is a very good starting point, but that's something that could have come from other sources than Christianity.