I think people confuse the fact that some of the country's earlier settlers arrived here so they could practice their own brand of Christianity with the ideas and intentions of the Founding Fathers. It has also become a common argument in modern political debate when justifying policies or candidates that follow common perceptions of Christian morality, as people can harken back to the Founding Fathers and what they "intended".
Unfortunately, I also find that most people know very little about the people who wrote our fouding documents at all, much less what they intended. As various sects of Christianity were already openly and widely practiced across what would become the 13 states, a cautious tolerance among the various faiths evolved into the 1st Amendment and freedom of religion. However, that amendment very carefully establishes separation of church and state, and while God and Christian beliefs were commonly held at the time, many if not most of the Founding Fathers were not overly pious men.
In short, they were creating a framework for democracy, not theocracy, and there is little evidence they intended to infuse religion permanently into the American legal system or government.
The Founders used their existing moral philosophies, based in Christianity and other religions, to create the framework of freedom in which the United States is set. The U.S. is not an explicitly Christian nation, but its founding principles came from Christian morals and ideals as interpreted by the Founders. Again, this does not mean that all the documents are Christian in nature, but that their writing -- like most writing of the time -- was inspired by those existing ideals.
The current-day push to remove all religious references, words, and ideas from lawful documents is one that would be alien to the people of the 1700s, simply because people tended to have religious beliefs and so would frame their ideas from that standpoint. In any case, the founding documents should never be censored, nor should any documents. The existence of religious language in the founding documents is a sign of their times, not of religious fundamentalism or oppression. Certainly, the documents have morals based in Judeo-Christian principles, and so do many, many other documents, books, films, and philosophies. Better to examine the substance and the issues at hand rather than focus on the religious influence.
Because it was an all Christian country at the time of the writing of the Constitution, the Founding Fathers perhaps did not feel the need of mentioning explicitly "God." However, as alreading mentioned, "God's nature" is written and the principles outlined in the document point to a culture that is Christian; after all, it was Puritans who arrived at Plymouth Rock and their communities were theocratic. In addition, the ideals of the U.S. Constitution were drawn from the thinking of John Locke and other Continental philosophers; therefore, Christian ideals are certainly inherent in this historical document.
But, as an important legal document, direct mention of God is not made since the federal government has held with the separation of Church and State.
There is certainly an implication of Christian ideology and moral and spiritual values in the founding documents, which can be seen in statements like this from the Declaration of Independence: "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."
The words truths, created, endowed, Creator, unalienable, and Rights, all imply the acknowledgement of precepts and teachings in Judeo-Christian theology. This is not unusual, though it may seem to be so today, because religion still dominated society in the 1700s, and the religion in the Colonies was various forms of Christianity.
Yet some of the Founding Fathers did not hold with Christianity personally, like Jefferson, though unable to escape the influence of Christianity on their belief systems and thoughts. Regardless of personal doubts or hesitations, the milieu, still Christian in tone and zeitgeist, molded the way the Founding Fathers thought and debated about Freedom and Liberty. This line of argument is supported by the fact that Jefferson's support base was Evangelical Baptists. Baylor University's Thomas S. Kidd's book God of Liberty: A Religious History of the American Revolution has much to say on this topic.
There was a wide range of religious belief among the politicians of the founding generation, and as the posts above have pointed out, the founding documents are not explicitly Christian. This was deliberate, of course, though there were many debates about the relationship between church and state that were going on at the state level. In Virginia, for example, Thomas Jefferson's Bill for Establishing Religious Freedom, which eliminated state support or favoritism toward any faith, met with opposition from a faction led by Patrick Henry, who favored a tax to support ministers. James Madison skillfully guided the bill through the Assembly after much debate. The point is that while most of the Founders were Christian, and imagined the United States as a Christian nation (though the meaning of that would have been highly contested) many understood religious freedom, which entailed for Jefferson a "wall of separation" between church and state, as essential for the health of religion in the country. Few debates about our history are, in my opinion, oversimplified by both sides as much as this one.
There is very little in the Constitution or the Declaration of Independence that is explicitly and exclusively Christian. This is not to say that the Founders were not influenced by Christianity. Instead, it is to say that they did not put anything in the founding documents that could not have been put there by a person of some other religion.
The Declaration of Independence does refer to God in a few places. However, it does so in very generic terms. It talks about things like the “supreme judge of the universe” and “nature’s God.” These are not things that are exclusively Christian, as they would be if the Founders had referred, for example, to the Trinity or Jesus Christ.
No one would deny that the Founders were heavily influenced by Christian values. But they did not explicitly insert Christianity into the founding documents.
It is true that the United States was mostly a country founded on Christianity. However important it was to the founders, they were also very careful to keep it out of the most important government document: The United States Constitution.
The U.S. Constitution establishes and defines the basic outlines of a national government that joins the states in an effective political union. (see first link)
The word "God" therefore does not appear anywhere in the document, and there is no mention of religion.
This lack of any these words does not mean that the Framers were not spiritual people, ...[it] is expositive of is not a love for or disdain for religion, but the feeling that the new government should not involve itself in matters of religion. (see second link)
The writers of the Constitution believed that the individual rights included in the Bill of Rights were important. However, they did not include them in the body of the Constitution. They were the first ten amendments as a compromise. The founders believed that the government should respect religion, but religion should not influence the government and government should not influence religion.