Independence Day, celebrated on July 4th in the United States, is not fundamentally religious in character. It is a secular holiday of comparatively recent origin. The holidays that Christians should not keep are those that are pagan in origin; but there is nothing inherently pagan about the 4th of July.
As a patriotic American citizen, there is certainly nothing wrong with observing a day that celebrates the freedoms bestowed upon this great nation by Almighty God. Patriotism becomes a sin only when it supersedes the worship of God!