John Adams wrote in a letter to his wife:
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 Shews, Games, Sports, Guns, Bells, Bonfires and Illuminations from one End of this Continent to the other from this Time forward forever more.
You will think me transported with Enthusiasm but I am not. — I am well aware of the Toil and Blood and Treasure, that it will cost Us to maintain this Declaration, and support and defend these States. — Yet through all the Gloom I can see the Rays of ravishing Light and Glory. I can see that the End is more than worth all the Means. And that Posterity will tryumph in that Days Transaction, even altho We should rue it, which I trust in God We shall not.
Prophetic, yes…but he was talking about the second of July, not the fourth. On July 2, the assembled representatives of the 13 colonies voted for independence. But it wasn’t until July 4 that the Declaration of Independence itself was presented for signing. And that’s the date that stuck.
If he could, he’d tell us we were doing it on the wrong day (and he did so every year while he lived), but leaving that error aside, we’ve been celebrating Independence Day according to Adams’ prescription for 242 years now.
Happy Birthday, America!
 Letter from John Adams to Abigail Adams, 3 July 1776, “Had a Declaration…” [electronic edition]. Adams Family Papers: An Electronic Archive. Massachusetts Historical Society. http://www.masshist.org/digitaladams/