The National Geographic sets the record straight:
Independence Day is celebrated two days too late. The Second Continental Congress voted for a Declaration of Independence on July 2, prompting John Adams to write his wife, "I am apt to believe that [July 2, 1776], will be celebrated, by succeeding Generations, as the great anniversary Festival."
Adams correctly foresaw shows, games, sports, buns, bells, and bonfires—but he got the date wrong. The written document wasn't edited and approved until the Fourth of July, and that was the date printers affixed to "broadside" announcements sent out across the land. July 2 was soon forgotten.
But, take heart. Signing of the Declaration of Independence didn't really begin until August 2nd, so you could legitimately celebrate Independence Day then. And, since the signing process wasn't finished until late November, you could legitimately celebrate Independence Day almost any day between August 2nd and December. Hey, you could stretch this Holiday out for months. What an idea.