July 4, 1776 is the date that the Second Continental Congress voted to adopt the Declaration of Independence. A vote had been held on July 2, 1776 to agree to seek independence from Britain in the form of the Lee Resolution. However, a formal Declaration of Independence did not appear until July 4th.
Perhaps the date we observe has as much to do with the stirring text of the Declaration as anything - and it is the formal document eventually signed by most of the delegates to the Congress.
While not a document which laid down the manner in which the government would be run, which would not arrive for over a decade in the form of the Constitution of the United States, the Declaration of Independence does lay down the moral reasoning for our separation from England. The Preamble, often memorized by school children over the years, and familiar to most Americans, formed the ethos of America as a state which required the consent of the governed, and that the government would serve the needs of the people. But also that government not be changed on a whim - but when the government no longer responded to the needs of the governed.