In reality, an Independence Day is an annual event commemorating the independence of a nation.
In the writer's first draft script of the Star Trek: The Original Series episode "The Squire of Gothos", Lieutenant DeSalle mentioned this concept in response to the USS Enterprise being greeted with a fireworks display upon arriving at Gothos, shortly after Captain Kirk and Lieutenant Sulu abruptly vanished from the starship's bridge. DeSalle specifically exclaimed, "Whose Independence Day is this supposed to be?" inferring that fireworks are often used to celebrate Independence Day. Yeoman Janice Rand joyfully replied, "Maybe Sulu's and the captain's–" although she was then cut off in midsentence by Spock. The mention of Independence Day was discarded from the story by the time the first draft of the teleplay was issued.