The moon is in free fall around the earth just like any other earth sattlelite, or like the cannonball fired from Newton's cannon. This was done by Henry Cavendish in the late 1700's. He created an ingenious device to directly measure the gravitational force between two pairs of lead balls. Three cheers for the brute force approach.
Gravity fills the universe.
Every particle of matter attracts every other through any distance just by virtue of being matter according to Newton's famous formula. Newton's 1st law explains why this is true. An object in motion continues in motion unless a net force acts upon it. Since it therefore does not act horizontally at all, it can't speed up the ball or slow it down in the horizontal direction. Thus the ball keeps moving at the same velocity horizontally even while it accelerates toward the ground.