  Emergence of classicality from quantum mechanics, a hotly debated topic, has
had no satisfactory resolution so far. Various approaches including decoherence
and gravitational interactions have been suggested. In the present work, the
Schr\"odinger-Newton model is used to study the role of semi-classical
self-gravity in the evolution of massive spin-1/2 particles in a Stern-Gerlach
experiment. For small mass, evolution of the initial wavepacket in a spin
superposition shows a splitting in the magnetic field gradient into two
trajectories as in the standard Stern-Gerlach experiment. For larger mass, the
deviations from the central path are less than in the standard Stern-Gerlach
case, while for high enough mass, the wavepacket does not split, and instead
follows the classical trajectory for a magnetic moment in inhomogeneous
magnetic field. This indicates the emergence of classicality due to
self-gravitational interaction when the mass is increased. In contrast,
decoherence which is a strong contender for emergence of classicality, leads to
a \emph{mixed state} of two trajectories corresponding to the spin-up and
spin-down states, and not the classically expected path. The classically
expected path of the particle probably cannot be explained even in the
many-worlds interpretation of quantum mechanics. Stern-Gerlach experiments in
the macroscopic domain are needed to settle this question.
