Yes, according to Newton's law of universal gravitation, the gravitational force exerted by Earth on the Moon is indeed the same as the gravitational force exerted by the Moon on Earth. This is known as Newton's third law of motion, often stated as "for every action, there is an equal and opposite reaction."
The law of universal gravitation states that every particle of matter attracts every other particle with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. In the case of Earth and the Moon, both objects exert gravitational forces on each other.
The force of gravity between two objects depends on their masses and the distance between them. While the mass of Earth is significantly greater than the mass of the Moon, the distance between the centers of the Earth and the Moon also plays a crucial role.
The gravitational force between Earth and the Moon causes the Moon to orbit around the Earth in a nearly circular path. Simultaneously, the Earth experiences a smaller but still significant gravitational force from the Moon. Without this mutual gravitational attraction, the Moon and Earth would not be in their current stable orbits.
So, while the masses of the two objects are different, the magnitudes of the gravitational forces they exert on each other are equal, but in opposite directions, as dictated by Newton's third law.