P: 991

Quote by Wikipedia
The Friedmann equations are a set of equations in cosmology that govern the expansion of space in homogeneous and isotropic models of the universe within the context of general relativity. They were first derived by Alexander Friedmann in 1922[1] from Einstein's field equations of gravitation for the FriedmannLemaîtreRobertsonWalker metric and a fluid with a given mass density ρ and pressure p. The equations for negative spatial curvature were given by Friedmann in 1924.

As stated by Wikipedia, the Friedmann equations are derived from General Relativity, not from Newtonian mechanics. If you are interested in how these equations were derived, I recommend reading Friedmann's actual papers themselves, they are listed in reference 1 below, at the end of page in Reference as well as most of the important equations.
Reference:
Friedmann equations  Wikipedia
