The constitution is a document dictating the way the country is governed....not what is is founded upon. It's a rule book for the country, it's not a history.
Here's the preamble to the Declaration of Independence---first sentence has God in it. Second sentence has "creator" in it.
http://www.archives.gov/exhibits/charters/declaration_transcript.html
"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."
Declaration of Independence was a testement to the reasons the country was declaring independence...not the constitution.
Considering that this nation is a nation of federated states...
http://www.usconstitution.net/states_god.html