We construct a homothetic covariant Newtonian gravitation theory which unifies inertial homothetic forces and gravitational fields. This is achieved through an equivalence principle based on a local homothetic frame of motion. As a consequence, we can obtain a coherent Newtonian cosmology which admits a cosmological principle and leads to the Friedman equations for a dust universe. Finally we prove that this gravity theory can be obtained as the non-relativistic limit of a class of metrics in General Relativity. The Friedmann–Lemaître–Robertson–Walker metric and its limit are also studied.