Neither Bastille Day or Independence Day marked the end of a revolution, but the beginning. In both cases, citizens rioted in an effort to inform their government they were unhappy and not willing to endure the injustices anymore. Both events are celebrated as a sort of patriotic individualism and the beginning of French and American culture today.
Of course, given the difference in circumstances and cultures, the two holidays can not be completely equated. However, the French were trying to throw off their ties to upper crust shackles at about the same time the United States was and the storming of the Bastille has the same symbolic value for them as the Fourth.
Bastille Day is definitely the equivalent to America's Independence Day. Described as "French National Day" this holiday celebrates all things French, and the rich history of the country, much like independence day recognizes the freedom of America, and the great country that we are blessed to live in. Both celebrations also tend to have great fireworks!
Bastille Day is celebrated on the 14th of July, and although they don't say they celebrate their Independence on this day, they do commemorate the beginning of their French Revolution. Which one can argue It was the beginning of their Independence. The evenings light up with fireworks and the morning is host to the oldest and largest regular military parade in Europe.