  We consider a two-Higgs doublet scenario containing three $SU(2)_L$ singlet
heavy neutrinos with Majorana masses. The second scalar doublet as well as the
neutrinos are odd under a $Z_2$ symmetry. This scenario not only generates
Majorana masses for the light neutrinos radiatively but also makes the lighter
of the neutral $Z_2$-odd scalars an eligible dark matter candidate, in addition
to triggering leptogenesis at the scale of the heavy neutrino masses. Taking
two representative values of this mass scale, we identify the allowed regions
of the parameter space of the model, which are consistent with all dark matter
constraints. At the same time, the running of quartic couplings in the scalar
potential to high scales is studied, thus subjecting the regions consistent
with dark matter constraints to further requirements of vacuum stability,
perturbativity and unitarity. It is found that part of the parameter space is
consistent with all of these requirements all the way up to the Planck scale,
and also yields the correct signal strength in the diphoton channel for the
scalar observed at the Large Hadron Collider.
