  For the one-dimensional mass-critical/supercritical pseudo-relativistic
nonlinear Schrodinger equation, a stationary solution can be constructed as an
energy minimizer under an additional kinetic energy constraint and the set of
energy minimizers is orbitally stable in \cite{BGV}. In this study, we proved
the local uniqueness and established the orbital stability of the solitary wave
by improving that of the energy minimizer set. A key aspect thereof is the
reformulation of the variational problem in the non-relativistic regime, which
we consider to be more natural because the proof extensively relies on the
subcritical nature of the limiting model. Thus, the role of the additional
constraint is clarified, a more suitable Gagliardo-Nirenberg inequality is
introduced, and the non-relativistic limit is proved. Subsequently, this limit
is employed to derive the local uniqueness and orbital stability.
