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Best Answer Larry Fields says
Yes, surely. You gave a great explanation and I fully agree. We can say that the Moon falls as well. Only one thing to add: the key difference between the free fall and the circular motion of the Moon is the initial velocity of a body.
Thanks duffsmom, but no... gravity doesn't need any media to act. According to the Newton's law of gravity, gravity act on both an apple and the Moon. Better to say, both of bodies interact with the Earth. The reason is in an another fact...