Why did Newton conclude that the gravitational force decreases with the square of the distance?
Newton observed that the centripetal acceleration of the Moon (am = V2Rm) was much smaller than the acceleration due to gravity on Earth’s surface (g), and their ratio matched the inverse square of their distances from Earth’s center (gam≈Rm2RE2).
Ref: Halliday & Resnick, Fundamentals of Physics, 11th ed., Chapter 13: Gravitation.