A resolution for independence was unanimously adopted by a roll call of the colonial delegations to the Continental Congress on July 2nd, following a recommendation for independence that had been approved by the Committee of the Whole (a procedural device consisting of all the delegates as if sitting in one large committee, used to ease strict general rules of parliamentary procedure that apply during the formal sitting of a legislature) the day before. On July 4th, the text of the Declaration of Independence (the first draft was written by Thomas Jefferson of Virginia, and then amended by the other members of the committee: John Adams of Massachusetts, Benjamin Franklin of Pennsylvania, Roger Sherman of Connecticut, and Robert Livingston of New York) was adopted after amendments by the Continental Congress (one of these was removal of a passage expressly denouncing slavery). The Declaration had been mostly prepared in advance in anticipation of a vote for independence, as the question had been moved by Richard Henry Lee of Virginia on June 7th and delayed until July 1st, partly to give the Declaration committee time to prepare a draft. It is believed by some historians that the engrossed Declaration copy that is in the National Archives wasn't signed until early August, and then in a piecemeal fashion. John Hancock, as President of the Continental Congress, almost certainly signed first.

TL; DR July 1: Committee of the Whole approves Lee Resolution for independence.

July 2: Continental Congress declares independence and the Declaration was introduced.

July 3: Declaration amended.

July 4: Declaration adopted.

Sources: This general timeline has been generally agreed to by historians. There are multiple letters from delegates to friends who corroborate the events of these few eventful days, as well as personal diaries. The Wikipedia page on the Declaration lists all of these sources in detail.

EDIT: Changed words and added the states from which the members of the Committee of Five hailed.

Ironically, the king of England wrote in his diary "Nothing of great importance happened on this day." It wasn't his way of being funny or bitter about America being independent, because news of total independence wouldn't reach his ears for another 3 months, as that's how long it took to cross the Atlantic via boat. This might not quite be the answer you were searching for, but it still happened on that day, and it's quite entertaining to know, IMHO.