In a letter to his wife Abigail from Philadelphia on July 3, 1776, he wrote,
The Second Day of July 1776, will be the most memorable Epocha, in the History of America.
His mistake is understandable, since all the signers of the Declaration of Independence didn't exactly gather around the parchment and sign all at once. For example, Adams wrote a letter to Samuel Chase of Maryland suggesting he add his name to the document. The letter was dated July 9, 1776.
Adams later said the Continental Congress signed the Declaration of Independence on July 4. So did Thomas Jefferson and Benjamin Franklin. But Adams' letter and additional evidence suggests otherwise. Some of the signers -- like Chase -- weren't in Philadelphia on July 4th. Moreover, the Declaration wasn't printed with all the names of the signers until January 1777, by Mary Katherine Goddard in Baltimore.
John Adams may have initially gotten the official date wrong, but he correctly predicted the anniversary celebrations that would follow. "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," he wrote to Abigail in that same letter. "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.