The moon instantly fell to the Ground, if it was fixed. But the Moon is not in place, it revolves around the Earth.
You can verify this yourself by conducting a simple experiment. Tie eraser thread and start to spin. The eraser on the thread will just break out of your hands, but the thread it will not be allowed. Now stop the rotation. Eraser immediately fall.
Even more illustrative example is the analogy - Ferris wheel. People don't fall out of this merry-go-round when they are at the highest point, despite the fact that they are upside down, because the centrifugal force, which throws them out (pulls to the seat)is greater than the gravitational force of the Earth. The speed of rotation of the Ferris wheel specially calculated, and if the centrifugal force is smaller than the force of Earth's gravity, it would end up a disaster - people would fall out of their cabins.
Exactly the same is the case with the Moon. The force that keeps the moon "escape" when the rotation is the force of attraction of the Earth. But the force that keeps the moon from falling on the Earth is the centrifugal force that occurs when the rotation of the moon around the Earth. Turning around the Earth, the Moon moves in an orbit with a speed of 1 km/sec, that is quite slow, so as not to leave its orbit and "fly" in space, but fast enough not to fall to the Ground.
By the way...
You will be surprised, but actually the Moon... moving away from earth at a speed of 3-4 cm per year! The movement of the moon around the Earth can be represented as a slowly unwinding spiral. The reason for this trajectory of the moon is the Sun, which attracts the moon 2 times stronger than Earth.
Why the Moon does not fall into the Sun? And because the Moon along with the Earth rotates, in turn, around the Sun, and the magnetic action of the Sun without a remainder is spent constantly to translate both of these bodies from the straight path on a curved orbit.