On July 4th 1776 some British subjects decided they didn’t want to be that anymore. They made the decision for all colonists even though the rest of the colonists didn’t really know about it and hadn’t elected them to do so. So a war happened.
Seven years later, England said “Fuck it, we’ve got better things to do.” The Treaty of Paris was signed on September 3rd, 1783.
So, in reality, it would be more accurate to call July 4th Declaration Day since all the future US did was say “Hey, we want to be independent.” Actual Independence Day was September 3rd, the day that the British admitted defeat…or, at least, said they just didn’t care enough about the North American colonies anymore.