A friend made the claim this weekend that were it not for the American Revolution, the French Revolution wouldn't have happened. After he said this, I realized that I had always taken this idea for granted, i.e. that the French Revolution was directly influenced by the American one, and that somehow they were necessarily linked. I think a teacher may have said something to that effect at some point in High School, and the idea was never really challenged again.
But it seems quite silly when I think of it now, as it assumes an American-centric role in history. Both were certainly liberal revolutions, but the French Revolution was a response to French problems, and the American Revolution to American problems.
Nonetheless, is there any truth to the idea that the French Revolution was directly influenced by the American one?
As a direct influence, no. The French Revolution, as you said, was a response of the people and nobility to gain control of the government to address problems of the French government, namely the debt. While there are minor influences, such as the Marquis de Lafayette having the Declaration of Independence near when he was writing the Declaration of the Rights of Man, the main influence was that of Rousseau.
The French Revolution was also very liberal with Rousseau was an ideological father whereas the American Revolution was much more conservative (with a focus on property and taxation) with the ideological father of John Locke. The two revolutions also had different aims, with a protection of property rights and freedom for the American Revolution whereas the French Revolution was focused on legal equality under the law, more extreme than the American experience.
Further, the idea that the American Revolution wouldn't have is too focused on American exceptionalism to be realistic. Outside of the economics of funding the American Revolution, it wasn't dependent on the American Revolution to inspire the people whom were more fueled by Aristocratic excess and the lack of rights than inspiration.