The Declaration of Independence was actually approved by the Second Continental Congress on 2 July 1776—the printed copy was not available until 4 July. At the time, John Adams predicted how Independence Day would be celebrated, relatively accurately, except he anticipated it would be celebrated on the second rather than the fourth.
“The second day of July 1776 will be the most memorable epoch in the history of America. I am apt to believe that it will be celebrated by succeeding generations as the great anniversary festival. It ought to be commemorated as the day of deliverance by solemn acts of devotion to God Almighty.
“It ought to be solemnized with pomp and parade, with shows, games, sports, guns, bells, bonfires, and illuminations, from one end of this continent to the other, from this time forward forever more.”
Unlike the famous portrait, not every delegate signed on 4 July. Some members were attending to other business and may have signed as late as September or October. Some members of the Continental Congress never signed, but most did, including at least one who had not been a member of the Second Continental Congress. (George Washington was not a member and did not sign–he was busy fighting the British.)
Ironically, John Adams and Thomas Jefferson both died 4 July 1826 —the fiftieth anniversary of the first signatures on the Declaration.