You answered your own question in the very first line. That is because the US declaration of Independence was ratified/approved on 4th of July. What you say is correct, the declaration itself was voted on by Congress on 2nd of July.
You are also correct that John Adams thought 2nd of July would be the most memorable day in American history as he wrote to his wife in a letter:
The Second Day of July 1776, will be the most memorable Epocha, in the
History of America.
While the resolution for independence was passed on 2nd July, it wasn't finalized until 4th of July. The Congress edited the existing proclamation, moderating it and doing grammatical edits before sending it to press in order to explain the move to the public. The five men committee responsible for that included John Adams, Roger Sherman, Robert Livingston, Benjamin Franklin and Thomas Jefferson. They took out extremist assertions which could harm American interests by alienating Sympathizers in British public (Such as Britain enforcing slavery on the Colonies), shortened the text, improved sentence structure etc. Then finally on 4th July, the act was finalized, adopted and sent to print.
So essentially, the answer is paperwork. Technically, the real Independence Day is 2nd of July however, It took the Congress two more days to adopt a finalized statement and send it to print. Which is why the date that appears on the declaration is 4th of July, not the second. See the original declaration below and notice the date:
This was used as such by the press, thereby making it imprint in mind of American populace that Independence day is 4th of July. The first official commemoration by continental Congress was also on 4th of July, making it the de-facto Independence day.
Then it was made formally de-jure when Congress passed a law in 1870 to make Independence Day a federal holiday, using the date 4th of July, thereby making it legal and official.