  We study the prospect of simultaneous explanation of tiny neutrino masses,
dark matter (DM), and the observed baryon asymmetry of the Universe in a
$Z_3$-symmetric complex singlet scalar extended type-II seesaw model. The
complex singlet scalar plays the role of DM. Analyzing the thermal history of
the model, we identify the region of the parameter space that can generate a
first-order electroweak phase transition (FOEWPT) in the early Universe, and
the resulting stochastic gravitational waves (GW) can be detected at future
space/ground-based GW experiments. First, we find that light triplet scalars do
favor an FOEWPT. In our study, we choose the type-II seesaw part of the
parameter space in such a way that light triplet scalars, especially the doubly
charged ones, evade the strong bounds from their canonical searches at the
Large Hadron Collider (LHC). However, the relevant part of the parameter space,
where FOEWPT can happen only due to strong SM doublet-triplet interactions, is
in tension with the SM-like Higgs decay to a pair of photons, which has already
excluded the bulk of this parameter space. On the other hand, the latest
spin-independent DM direct detection constraints from XENON-1T and PANDA-4T
eliminate a significant amount of parameter space relevant for the dark sector
assisted FOEWPT scenarios, and it is only possible when the complex scalar DM
is significantly underabundant. In short, we conclude from our analysis that
the absence of new physics at the HL-LHC and/or various DM experiments in the
near future will severely limit the prospects of detecting a stochastic GW at
future GW experiments and will exclude the possibility of electroweak
baryogenesis within this model.
