"We hold these truths to be self-evident, that all men are created equal..." 
Religion's True Part in Building the United States
Let's address some misconceptions first.
First off, we need to address the idea that all the original colonists were Puritan religious refugees. This simply isn't true. Like the United States today, the original settlers were a hodgepodge of various cultural and religious beliefs. Now, it is true that Puritans did make up the majority, and it is also true that most communities or colonial cities were built around predominantly Christian populations. But there were still others who simply came here for a shot at a better life. They saw a place where land was plentiful and a man might do well for himself. However small their part and contribution, they are still a part of the background of this nation.
The second thing to clear up is that the original colonists are not representative of the populous during the Revolutionary War. By that point, the diversity of culture and beliefs in this unformed nation had grown tremendously. Even Christianity had introduced many new denominations and become far more divided. These were not merely colonists, but rather citizens in proud and thriving metropolises of their time. They had built these cities up from frail wood structures to ones with real permanence made of brick and mortar. As such, most of them didn't even identify as British or colonists at all.
So what role did religion play?
To figure that out, we must look to the British government of that time. You see, although that government was a constitutional monarchy, it was a direct descendent of a theocratic monarchy, and religion was still highly ingrained in the government. Because of this, there was general discrimination of any ideas outside the theocratic norm. With such a strong influence in parliament, the religious element in Britain was still basically completely in control. Their main goal, just as the king's, was to amass greater wealth; and what better way to do that than by taxation.
Dissolving the Ties that Bind
So, with this in mind, we should understand that the men who founded the US were not Puritan or other denominational religious zealots. They had no wish to set up a new monarchy or a new theocracy. If we just read the actual declaration we can see exactly what they wished to set up.
"When in the Course of human events, it becomes necessary for one people to dissolve the political bands which have connected them with another, and to assume among the powers of the earth, the separate and equal station to which the Laws of Nature and of Nature's God entitle them, a decent respect to the opinions of mankind requires that they should declare the causes which impel them to the separation.
We hold these truths to be self-evident, that all men are created equal, that they are endowed by their Creator with certain unalienable Rights, that among these are Life, Liberty and the pursuit of Happiness.--That to secure these rights, Governments are instituted among Men, deriving their just powers from the consent of the governed, --That whenever any Form of Government becomes destructive of these ends, it is the Right of the People to alter or to abolish it, and to institute new Government, laying its foundation on such principles and organizing its powers in such form, as to them shall seem most likely to effect their Safety and Happiness. Prudence, indeed, will dictate that Governments long established should not be changed for light and transient causes; and accordingly all experience hath shewn, that mankind are more disposed to suffer, while evils are sufferable, than to right themselves by abolishing the forms to which they are accustomed. But when a long train of abuses and usurpations, pursuing invariably the same Object evinces a design to reduce them under absolute Despotism, it is their right, it is their duty, to throw off such Government, and to provide new Guards for their future security.--Such has been the patient sufferance of these Colonies; and such is now the necessity which constrains them to alter their former Systems of Government. The history of the present King of Great Britain is a history of repeated injuries and usurpations, all having in direct object the establishment of an absolute Tyranny over these States." 
These are the first two paragraphs of the Declaration of Independance, and I want you to notice something. Twice, our founders mention a god and a creator. For Christians, this is supposedly a sign of their endorsement of Christianity. But I ask you to look closely, because I see no endorsement of Christ or Christianity here. What I see is a framing that endorses the beliefs of all men. I see no endorsement of the Bible or Jesus or Yahweh. All I see is a mention of "Nature's God" and a "creator." We well know that some of these men were indeed Christians, as well as deists, and possibly even a few quiet atheists. And it is because of this, because of this diversity amongst founders, that we see an open writ that might apply to all men. While I may not subscribe to the notion of a god of nature, I was indeed created. My creators are my parents, and by my estimation this means the very act of being born, of existing, endows me with certain rights. Not only in this nation, but as a human being. That's why we call them human rights, because one only needs be a human for those rights to apply.
The Establishment Clause
Our founders sought as much to unburden themselves of the shackles of oppressive religious groups like the Catholic church and the Church of England, as they did the oppression of a tyrannical king. It wasn't merely taxation without representation which drove them, but rather being suffocated by an oppressive regime which included many religious organizations. The religious world had so infiltrated the British government that there was little to no true separation. Our founders saw this, and saw how it caused injustice and disparity and wished to build a system free of this influence. It was so important to them that the very first amendment to the constitution, also known as the Establishment Clause, deals with it directly.
"Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof; or abridging the freedom of speech, or of the press; or the right of the people peaceably to assemble, and to petition the government for a redress of grievances." 
The United States is not a Christian nation and it never has been. Our founders risked all they had, including their very lives, to establish a nation free to all men regardless of religion. We've made great strides to go beyond that simple idea and say that freedom in this nation must apply to each and every citizen regardless of age, race, gender, or ethnicity. But we have further yet to go, because as long as even one person is denied equality and freedom in this nation, none of us are truly free. How can I enjoy my freedom when my neighbors are denied the same basic rights as I am?
This progress we need to make has one obstacle to overcome, and that is Christian fundamentalism. These religious fanatics are hell-bent on turning a vibrant, secular nation into a theocratic prison. Sadly, they've infiltrated our government deeply and stemming the tide is going to take a great deal of very hard work.
No.... The United States is not a Christian nation. It is a nation that has been taken hostage by Christian terrorists.