  A phenomenological anisotropic variation \Delta G/G of the Newtonian
gravitational coupling parameter G, if real, would affect the orbital dynamics
of a two-body gravitationally bound system in a specific way. We analytically
work out the long-term effects that such a putative modification of the usual
Newtonian inverse-square law would induce on the trajectory of a test particle
orbiting a central mass. Without making any a-priori simplifying assumptions
concerning the orbital configuration of the test particle, it turns out that
its osculating semi-major axis a, eccentricity e, pericenter \varpi and mean
anomaly M undergo long-term temporal variations, while the inclination I and
the node \Omega are left unaffected. Moreover, the radial and the transverse
components of the position and the velocity vectors r and v of the test
particle experience non-vanishing changes per orbit, contrary to the
out-of-plane ones. Then, we compute our theoretical predictions for some of the
major bodies of the solar system by orienting the gradient of G(r) towards the
Galactic Center and keeping it fixed over the characteristic timescales
involved. By comparing our calculation to the latest observational
determinations for the same bodies, we infer \Delta G/G <= 10^-17 over about 1
au. Finally, we consider also the Supermassive Black Hole hosted by the
Galactic Center in Sgr A^* and the main sequence star S2 orbiting it in about
16 yr, obtaining just \Delta G/G <= 10^-2 over 1 kau.
