  We consider a quantum scalar field on an arbitrary gravitational background.
We obtain the effective {\it in-in} equations for the gravitational fields
using a covariant and non-local approximation for the effective action proposed
by Vilkovisky and collaborators. {}From these equations, we compute the quantum
corrections to the Newtonian potential. We find logarithmic corrections which
we identify as the running of the gravitational constants. This running
coincides with the renormalization group prediction only for minimal and
conformal coupling.
