A planet following this law would either:
Follow a circular orbit. In this case the value of R would be constant and so the magnitude of the central force would also be constant, only the direction would change. Any central force permits circular orbits, and circular orbits have eccentricity zero. This kind of orbit has a zero probably of occurring if the initial velocity and separation are chosen randomly
Not have a closed orbit. Here is a picture of a bit of such an orbit:
(generated by code adapted by Zack Fizell
As such an orbit isn't an ellipse, the eccentricity is not defined. Such an orbit has probably 1 of occurring if the initial conditions are random.
Your question cannot be answered. The eccentricity or more generally the shape of an orbit is not completely determined by the central force. Even with Newtonian gravity, you can't determine the eccentricity only from the law of gravitation.