It's more accurate than it might sound. If I recall correctly, the American spelling reforms were largely a one-man top-down effort driven by the ideological goal of being more separate from Britain.
You can say any date like that in American English, it’s just less common. “Today is the second of February” sounds perfectly natural, but I’d be more likely to say “Today is February second”.
And a proper name is not good evidence of normal usage. "Christmas" is normal use in an older, Catholic method of identifying calendar dates. Americans don't use that method; most couldn't if they wanted to. But Christmas itself is still identified that way for historical reasons -- the word has gone from being an example of talking about the calendar to being an example of a proper name.
"4th of July" is not a method that Americans couldn't use if they wanted to, but it is an ossified usage that doesn't reflect how Americans generally refer to dates. Like Christmas, it is a proper name of limited evidentiary value.
Possibly it's the difference between being an atheist and a catholic, since I faintly remember my catholic friends receiving presents on the morning of the 25th, while I would receive them on the evening of the 24th.
