  The scope of this paper is twofold. First, we derive rigorously a
low-velocity and Galilei-covariant limit of the gravitoelectromagnetic (GEM)
equations. Subsequently, these reduced GEM equations are coupled to the
Schr\"odinger equation with gravitoelectric and gravitomagnetic potentials. The
resulting extended Schr\"odinger-Newton equations constitute a minimal model
where the three fundamental constants of nature ($G$, $\hbar$, and $c$) appear
naturally. We show that the relativistic correction coming from the
gravitomagnetic potential scales as the ratio of the mass of the system to the
Planck mass, and that it reinforces the standard Newtonian (gravitoelectric)
attraction. The theory is further generalized to many particles through a
Wigner function approach.
