You forget, as it appears most politicians and most people today also forget, that one of our founding documents does in fact recognize God explicitly.
The Constitution was written by Deists, not Catholics, or Protestants, and the god they mentioned, and this only in the Declaration of Independence, was Locke's "god of nature." They did not want the Republic to profess the Christian religion, where it would profess the Trinity and declare Jesus Christ to be the King of this nation, or any religion at all.
There was a group, the National Reform Association, that lamented the Deistic Constitution that made the U.S. entirely unconfessional, and from reviewing the history and the frank admissions of the Founders that the nation was founded on godless Enlightenment and not Protestant Christian principles. The National Reform Association saw this fact as the key flaw to the Constitution that lead to the War Between the States.
One such admission by the generation of the Founders lies in the declaration of the U.S. Senate in Article XI of the Treaty of Tripoli, approved by John Adams on June 10, 1797, that:
As the Government of the United States of America is not, in any sense, founded on the Christian religion, as it has in itself no character of enmity against the laws, religion, or tranquility of Mussulmen (Moslems) ... it is declared by the parties, that no pretext arising from religious opinions, shall ever produce an interruption of the harmony existing between the two countries.
The National Reform Association sought to add an amendment to the Preamble of the Constitution confessing Christ the King of the nations. Their proposed rewrite went as follows:
"We the People of the United States, [humbly acknowledging Almighty God as the source of all authority and power in civil government, the Lord Jesus Christ as the Ruler among the nations, His revealed will as the supreme law of the land, in order to constitute a Christian government, and] in order to form a more perfect union..."[Proceedings of the NRA, 1872]
This was tabled by Congress in 1874. In the report of the House Judiciary Committee that rejected this out of hand on February 18, 1874, it was declared that, after due examination of the record of the debates at the Constitutional Convention, it was revealed that:
The fathers of the Republic in the convention which framed the Constitution...decided that this country, the foundation of whose government they were laying, was to be the home of the oppressed of all the nations of the Earth, whether Christian or Pagan, and in full realization of the dangers which the union between Church and state had imposed upon so many nations of the Old World, [decided] with great unanimity that it was inexpedient to put anything in the Constitution or frame of government which might be construed to be a reference to any religious creed or doctrine.["House Reports," Vol I, 43rd Congress, 1st Session, Report No. 143, cited in Liberty, the God that Failed, C. Ferrara]