  We investigate a minimal $U(1)'$ extension of the Standard Model with one
extra complex scalar and generic gauge charge assignments. We use a type-I
seesaw mechanism with three heavy right handed neutrinos to illustrate the
constraints on the charges, on their mass and on the mixing angle of the two
scalars, derived by requiring the vacuum stability of the scalar potential. We
focus our study on a scenario which could be accessible at the LHC, by
selecting a vacuum expectation value of the extra Higgs in the TeV range and
determining the constraints that emerge in the parameter space. To illustrate
the generality of the approach, specific gauge choices corresponding to
$U(1)_{B-L}, U(1)_R$ and $U(1)_\chi$ are separately analyzed. Our results are
based on a modified expression of one of the $\beta$ functions of the quartic
couplings of the scalar potential compared to the previous literature. This is
due to a change in the coefficient of the Yukawa term of the right handed
neutrinos. Differently from previous analysis, we show that this coupling may
destabilize the vacuum.
