The short answer is yes, the Fourth of July is capitalized because it is a special date, a holiday. You should capitalize the “Fourth” and “July,” but lowercase the “of” because “of” is a short word.
If you are referring to the Fourth as “Independence Day” then it should also be capitalized since again it is a special date.
The other thing to keep in mind when spelling out “Fourth of July” is that you should not shorten “Fourth” to “4th.” It turns out various style guides have rules that apply for times when numbers are spelled out. Let’s take a look:
1. When talking about age (for people and animals), numerals are used.
E.g. Jason is 17 years old.
2. Single digit numbers (less than 10) should be spelled out.
E.g. The American Flag has 50 stars, but only three colors. On average, Americans have four beers during Independence Day celebrations.
3. Ordinal numbers below 10th need to be spelled out. Any number over that is written down in numerals
E.g The Fourth of July Celebrations just began.
The third rule is especially important. We can always say Fourth of July, July Fourth or July 4. It is however grammatically incorrect to say 4th of July.