  In a class of gauged $U(1)$ extended Standard Models (SMs), the breaking of
the $U(1)$ symmetry is not only a source for Majorana masses of right-handed
(RH) neutrinos crucial for the seesaw mechanism, but also a source of
stochastic gravitational wave (GW) background. Such $U(1)$ extended models are
well-motivated from the viewpoint of grand unification. In this paper, we
discuss a successful ultraviolet completion of a $U(1)$ extended SM by an
$SO(10)$ grand unified model through an intermediate step of $SU(5) \times
U(1)$ unification. With a parameter set that is compatible with the $SO(10)$
grand unification, we find that a first-order phase transition associated with
the $U(1)$ symmetry breaking can be strong enough to generate GWs with a
detectable size of amplitude. We also find that the resultant GW amplitude
reduces and its peak frequency becomes higher as the RH neutrino masses
increase.
