We the People of the United States .... do ordain and establish this Constitution for the United States of America.
The
verb in the preamble of the Constituion is 'do'. The proper noun in the preamble is 'People' (defining the subject of the sentence, the pronoun 'We'). The
People do establish the Constitution, Not God.
The purpose of the Constitution is to outline what role Government is supposed to take in the peoples lives: 'to create a more perfect union, establish justice, insure domestic tranquility, provide for the common defense, promote the general welfare, and secure the blessings of liberty for ourselves and our posterity'.
The quoatation you site from the
Declaration of Independence references 'Nature's God' and 'Creator'. But by leaving out the next sentence of you miss the whole point.
That to secure these rights, Governments are instituted among Men, deriving their just powers from the consent of the governed,
Governments are instutitued among Men. Governments derive their power from the consent of the governed. So, while Nature's God may have a hand in 'Life, Liberty, and the Pursuit of Happiness', It plays little role in the creation of a government.
Notes:
Declaration of Independence signed : July 4, 1776
United States Constitution ratified : June 21, 1788
There is a big time difference between these to documents.