We say the fourth of July. Because we've added the article, we default back to the normal grammar in our shared language of putting the an adjective before a noun. Makes you wonder why we otherwise have the calendar convention -- but it is what it is. If you look at a calendar, a date is something you can literally point to. So it is easy for our brain to treat it as a noun -- or rather the combination of this fact and the cultural conditioning that is having a language makes us used to parsing it that way. But the holiday itself, is not the calendar item. So it's either "the July 4th holiday" or "the 4th of July" -- and people are almost always going to take the easier route.

Just thinking about how we use it, we mainly flex into calling it the 4th of July when we are talking about it in the capacity of a holiday. An American putting it on the calendar could say something like "what are our plans for July 4th?"

Contrast this with when we talk about it as a holiday, like "what is your favorite food for the fourth of July?"