The door keeps you from sliding out of the car. It applies a centripetal force directed towards the turning point which keeps you from flying out onto the road. For the Moon its Gravity which supplies the force which produces the centripetal acceleration. In a sense the Moon is always falling towards the Earth, it just keeps missing because of its horizontal motion. That is a pretty small number.
Gravity is a pretty weak force as fundmental forces go.
If m E is the Earth and m m is the moon, Newton's second law for the Moon becomes Notice how the moon's mass goes away in the last equation. That is why Galileo was right about heavy and light falling bodies. They both fall with the same acceleration. Lets look at the apple and the moon problem. Newton knew Earth-Moon distance was about 60 Earth radii.