  We derive the non-relativistic limit of a massive vector field. We show that
the Cartesian spatial components of the vector behave as three identical,
non-interacting scalar fields. We find classes of spherical, cylindrical, and
planar self-gravitating vector solitons in the Newtonian limit. The
gravitational properties of the lowest-energy vector solitons$\mathrm{-}$the
gravitational potential and density field$\mathrm{-}$depend only on the net
mass of the soliton and the vector particle mass. In particular, these
self-gravitating, ground-state vector solitons are independent of the
distribution of energy across the vector field components, and are
indistinguishable from their scalar-field counterparts. Fuzzy Vector Dark
Matter models can therefore give rise to halo cores with identical
observational properties to the ones in scalar Fuzzy Dark Matter models. We
also provide novel hedgehog vector soliton solutions, which cannot be observed
in scalar-field theories. The gravitational binding of the lowest-energy
hedgehog halo is about three times weaker than the ground-state vector soliton.
Finally, we show that no spherically symmetric solitons exist with a
divergence-free vector field.
