  If neutrino masses arise from a TeV-scale minimal Left-Right seesaw model,
the ensuing extended Higgs sector with neutral, singly and doubly-charged
scalars has a plethora of implications for new Higgs boson searches beyond the
Standard Model at future hadron colliders, such as the $\sqrt s=14$ TeV
High-Luminosity Large Hadron Collider (HL-LHC) and the proposed $\sqrt s=100$
TeV collider (FCC-hh or SPPC). In this article, we provide a glimpse of this
new physics in the Higgs sector. Our discussion focuses on the minimal
non-supersymmetric version of the Left-Right model with high-scale parity
breaking but TeV-scale $SU(2)_R$-breaking, a property desirable to suppress the
type-II seesaw contribution to neutrino masses. We analyze the masses and
couplings of the physical Higgs bosons in this model, and discuss their
dominant production and decay modes at hadron colliders. We identify the best
discovery channels for each of the non-SM Higgs bosons and estimate the
expected SM backgrounds in these channels to derive the sensitivity reaches for
the new Higgs sector at future hadron colliders under discussion. Following a
rather conservative approach, we estimate that the heavy Higgs sector can be
effectively probed up to 15 TeV at the $\sqrt s=100$ TeV machine. We also
discuss how the LR Higgs sector can be distinguished from other extended Higgs
sectors.
