Not Newton but Gauss

Newton only had a vague idea of gravity. The first scientist to derive the universal gravitational law Fg = G*(M*m/sma^2) was Gauss. He derived it from Kepler's planetary laws. First Gauss established the value of G (6.674E-11) with G = sma*v^2/M where sma = semi major axis in m of any planet, v = periodical velocity in m of the same planet and M = the mass of the sun. The gravitational constant G was found in 1798 by Cavendish in the laboratory but Gauss calculated it with his own formula.

Gauss then used two laws to formulate Fg = G*(M*m/sma^2):
The area swept by a line joining a body and the Sun divided by the time in which it is swept gives a constant quotient. This is Kepler's second law of planetary motion.
The square of this quotient is proportional to the parameter p (that is, the latus rectum) of the orbit and the sum of the mass of the Sun and the orbiting body. This is a modified form of Kepler's third law.