In fact, the 3rd and 4th of July were taken up with debate and amendments to the 1,338-word explanation written by Thomas Jefferson and a small committee. A vote approving the text was taken on the 4th, but the actual parchment was not signed until August 2. That document used the July 4 date, and we've been stuck with it ever since.
There's another reason for celebrating July 2 -- for it was on that date, eleven years later in the same building in Philadelphia, that the Constitutional Convention broke its deadlock over how to organize the new government. Virginia had proposed proportional representation by population, including slaves. The small states, led by New Jersey proposed equal representation of the states in Congress. The debate and defeat of various plans left many small state delegates angry and frustrated, and drove the convention to the verge of collapse. Many considered leaving the convention if their rights were not protected. Delegates on both sides became more heated and intransigent.
The small states had a point. The three largest states of Virginia, Pennsylvania and Massachusetts had 45% of the U.S. population and would need only one other state to have a working majority over all other states. On many issues the three most southern states – Georgia and the Carolinas – sided with the big three. Though small at the time, they all expected to grow much larger and saw such an informal alliance as helpful to their other interests.
On July 2, 1787, Oliver Ellsworth of Connecticut pressed his resolution for equal representation in the Senate, but with some absentees and some still unexplained vote switches, the delegates tied on the question. That was a parliamentary defeat for the small states, but a game-changing, emotional victory because it kept alive their alternative. On reflection, other delegates realized that this issue of Senate composition could destroy any chance at government reform. They agreed to turn the question over to a committee, and three days later, the committee recommended equal votes in the Senate. At the end of the tumultuous week, the delegates approved the plan.
This broke the logjam on other issues as well. With small state rights protected, their delegates were more willing to strengthen the executive and the central government. By mid-July, delegates agreed on a single executive and gave him veto power.
Those are two strong reasons for venerating and celebrating July 2 -- with "bonfires and illuminations" and good beer.