There are lots of days to choose from that could be representative of American Independence, but John Adams (who really knows best, I would think) was planning on our celebrating the day on July 2, not 4. The 2nd makes more sense as a momentous day, as it was when the Second Continental Congress convened in Philadelphia to vote on a resolution of independence from Britain.
July 4 is when the Congress adopted Jefferson's Declaration of Independence, but it wasn't signed then. There was no grand convention of guys applying their signature all at once; they just trickled in whenever, and most didn't sign until around August 2.
Right away, Adams had ideas about how the day should be celebrated, and naturally he told his wife all about it, writing two letters in one day.
July 3, 1776, AM:
"Yesterday the greatest question was decided, which ever was debated in America, and a greater, perhaps, never was or will be decided among Men. A resolution was passed without one dissenting colony "that these United Colonies are, and of right ought to be, free and independent states, and as such they have, and of right ought to have, full power to make war, conclude peace, establish commerce, and to do all the other acts and things which other states may rightfully do." You will see in a few days a declaration setting forth the causes which have impelled us to this mighty revolution and the reasons which will justify it in the sight of God and man. A plan of confederation will be taken up in a few days."
July 3, 1776, PM:
"The second day of July, 1776, will be memorable epocha 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, shews, games, sports, guns, bells, bonfires and illuminations, from one end of the continent to the other, from this time forward forever."
John; would you settle for hot dogs and blackouts?
Nevertheless, the 4th is the day that was adopted, because celebrating the Declaration apparently seemed like it had more gravity than the day on which the Congress was together and cast a unanimous vote to throw off their parent country. I disagree, obviously. But then, 50 years later, former and recently reconciled bffs John Adams and Thomas Jefferson managed to die on the same day, within hours of each other on July 4, 1826. The coincidence is so strong that it seems quite relevant. July 4 it is.
See here for the Massachusetts Historical Society's massive collection of Adams letters.
And here is some exciting and appropriate music for the holiday.