  The $\mu\nu$SSM is a simple supersymmetric extension of the Standard Model
(SM) capable of predicting neutrino physics in agreement with experiment. In
this paper we perform the complete one-loop renormalization of the neutral
scalar sector of the $\mu\nu$SSM with one generation of right-handed neutrinos
in a mixed on-shell/$\overline{\mathrm{DR}}$ scheme. The renormalization
procedure is discussed in detail, emphasizing conceptual differences to the
minimal (MSSM) and next-to-minimal (NMSSM) supersymmetric standard model
regarding the field renormalization and the treatment of non-flavor-diagonal
soft mass parameters, which have their origin in the breaking of $R$-parity in
the $\mu\nu$SSM. We calculate the full one-loop corrections to the neutral
scalar masses of the $\mu\nu$SSM. The one-loop contributions are supplemented
by available MSSM higher-order corrections. We obtain numerical results for a
SM-like Higgs boson mass consistent with experimental bounds. We compare our
results to predictions in the NMSSM to obtain a measure for the significance of
genuine $\mu\nu$SSM-like contributions. We only find minor corrections due to
the smallness of the neutrino Yukawa couplings, indicating that the Higgs boson
mass calculations in the $\mu\nu$SSM are at the same level of accuracy as in
the NMSSM. Finally we show that the $\mu\nu$SSM can accomodate a Higgs boson
that could explain an excess of $\gamma\gamma$ events at $\sim 96\,
\mathrm{GeV}$ as reported by CMS, as well as the $2\,\sigma$ excess of $b
\bar{b}$ events observed at LEP at a similar mass scale.
