Was the American War of Independence a major player in the French Revolution of 1789?

This is a difficult question to clearly answer because it requires you to differentiate between the influences that the American Revolution had on the French Revolution and the influences that were shared between the two: Enlightenment thinking played a role in framing the issues in both revolutions. In both cases, economic policy that benefitted only wealthy aristocrats led to discontent among the impoverished, a sentiment that was largely redirected towards the monarchy; economic issues were blamed on the monarch, leading to significant dissent with this form of government among the revolutionaries.
There are two areas where it is clear that the American Revolution played a role in influencing the French Revolution.
In the first instance, the economic issues within France were dire because of the monarchy's lavish spending. However, a significant portion of this spending had been directed at weakening France's main rival: England. France had spent a significant amount of money to fund the American Revolution in order to weaken England. This included the provision of uniforms, weapons, ammunition, soldiers, and naval reinforcements. France—along with European allies—took action against English naval vessels in order to prevent the resupply and reinforcement of English soldiers in the colonies. Thus, France's economic situation (which eventually spurred the uprising) was heavily influenced by their involvement in the American Revolution.
In the second case, the American Revolution and the Declaration of Independence set an example for successfully overthrowing monarchical tyranny and establishing a democratic form of government.
Determining the importance of the American Revolution in influencing the French Revolution requires weighing these two issues, where the influence is clear and direct, against other issues that were influenced by shared Enlightenment ideologies as well as issues unique to France.
Interestingly, if France had not intervened to aid the colonists, the French economic situation may have been tenable, and the colonists would likely have lost the war for independence. The absence of a revolutionary precedent, combined with a better economic situation, might have significantly delayed or even eliminated the possibility of French revolution.