It’s hardly worth noting (since noone actually cares) that the actual birth date of the United States of America (as such) is September 17, 1787. The anniversary of this date is called “Constitution Day” and noone actually seems to care.
The Declaration of Independence was voted out by a body called the Continental Congress, an ad-hoc alliance of thirteeen British colonies in the Americas that had just entered into open rebellion against parliament and the crown. It is this monumental act of defiance that we celebrate with fireworks, family picnics, and a day off of work.
These American colonies would not gain their independence in fact until September 3, 1783 when the Treaty of Paris ended the American Revolutionary War.