We apply the Ricci flow to quantum gravity as a diffeomorphism invariant UV regulator for composite operators. Following steps analogous to the gradient flow in QCD, the Ricci flow equation is solved perturbatively by introducing flowed graviton propagators and vertices as well as graviton flow lines which describe the evolution of gravity along the Ricci flow. By calculating vacuum expectation values of a number of independent operators at the two-loop level, we derive the required counterterms of the flowed action. Our results allow us to define a Ricci-flow based renormalization scheme for Newton’s constant $g_N$. Studying its renormalization group behavior, we recover a non-Gaußian fixed point in a perturbative regime, without relying on any non-perturbative input.