Question

why the moon in space not fall on the earth even while earth has gravity it could not attract the earth

Answer 1

Because Earth rotates itself.The energy from the Earth's own rotation around its axis is  transferred into energy of the Moon's orbital motion. That's why the Earth's rotating speed decreases but the distance to the Moon increases.This process will continue until Earth's proper rotation will slow to the point where it has the same angular velocity as the Moon's orbital motion. From that moment on the Moon will start to gradually approach Earth.

Answer 2

yes the earth also attracts the moon according to the third law of motion every object has an equal and opposite reaction but according to the second law of motion acceleration produce per our earth so small due to its large mass that it cannot be noticed; this is the reason for the same