  The Schr\"odinger-Newton model is a semi-classical theory in which, in
addition to mutual attraction, massive quantum particles interact with their
own gravitational fields. While there are many studies on the phenomenology of
single particles, correlation dynamics in multipartite systems is largely
unexplored. Here, we show that the Schr\"odinger-Newton interactions preserve
the product form of initial states, yet on average it agrees with classical
mechanics of continuous mass distributions. This leads to a simple test of the
model, based on verifying bipartite gravitational evolution towards non-product
states. We show using standard quantum mechanics that, with currently
accessible single-particle parameters, two masses released from harmonic traps
get correlated well before any observable entanglement is accumulated.
Therefore, the Schr\"odinger-Newton model can be tested with setups aimed at
observation of gravitational entanglement with significantly relaxed
requirements on coherence time. We also present a mixed-state extension of the
model that avoids superluminal signaling.
