  It has recently been demonstrated [A. Giusti, Phys. Rev. D 101, 124029
(2020)] that characteristic traits of Milgrom's modified Newtonian dynamics
(MOND) can be replicated from an entirely distinct framework: a fractional
variant of Newtonian mechanics. To further assess its validity, this proposal
needs to be tested in relevant astrophysical scenarios. Here, we investigate
its implications on Jeans gravitational instability and related phenomena. We
examine scenarios involving classical matter confined by gravity and extend our
analysis to the quantum domain, through a Schr\"odinger-Newton approach. We
also derive a generalized Lane-Emden equation associated with fractional
gravity. Through comparisons between the derived stability criteria and the
observed stability of Bok globules, we establish constraints on the theory's
parameters to align with observational data.
