Greek mythology does not have as much of an impact on American culture as it has on Western culture. American culture is, strangely, more impacted by Norse Mythology than it is by its older counterpart, but that does not mean that Greek culture has had no impact. The Greek and Norse mythologies are often taught in schools. The impact on our culture is undeniable, but it goes further than just Greek mythology. A statue to a Roan goddess guards the entrance to New York.
It is often hard, especially during the Modern era, to tell the myths that were designed to entertain apart from the myths that were designed to tell deeper lessons. Many of the myths that children learn in schools are designed to entertain. Many of the transformation myths were the creation of the Roman writer Ovid and not designed to teach anything.
Of course, oddly enough, the Greeks of the golden age took many of the same attitudes we do today towards the myths. They did not believe that Apollo literally pulled a chariot across the sky. They understood the rotation of the Earth and many other things. The attitudes of the Greeks led to an early development of Science.
It could be better said that the idea of the Greeks had far more of an impact on American and Western culture than their myths did. There are other stories that are true that have inspired the west for many years. The story of the Battle of Marathon, and the story of the 300 Spartans who defended the pass at Thermopylae. Unfortunately, when Hollywood decides to retell the stories they focus on the special effects rather than the actual story telling. The most recent remake of Clash of the Titans was a good example of Hollywood ignoring the story of Perseus and confusing several different myths.