How did Newton show that the same gravity affects falling objects and orbiting bodies?

Isaac Newton demonstrated that the same gravitational force affects both falling objects on Earth and orbiting bodies by applying his laws of motion to compare their respective accelerations.12

Newton realized that a body in motion would continue in a straight line unless acted upon by a force, meaning a constant inward force is required to keep the Moon in its circular orbit rather than moving in a straight line. By analyzing Kepler’s third law, he determined that this centripetal force must decrease with the square of the distance from the center of motion.132

To prove this force was the same as terrestrial gravity, Newton compared the distance the Moon is diverted from a tangential path in one second with the distance a body at Earth's surface falls from rest in one second. He found that the Moon’s inward acceleration is approximately $1/3,600$ of the acceleration of a falling object at Earth's surface. Because the Moon is about 60 Earth radii away, and ${60}^2=3,600$, he concluded that gravity follows an inverse-square law, acting identically on both the Moon and falling objects.21