  We consider the minimal U(1)$_{B-L}$ extension of the Standard Model (SM)
with the classically conformal invariance, where an anomaly free U(1)$_{B-L}$
gauge symmetry is introduced along with three generations of right-handed
neutrinos and a U(1)$_{B-L}$ Higgs field. Because of the classically conformal
symmetry, all dimensional parameters are forbidden. The $B-L$ gauge symmetry is
radiatively broken through the Coleman-Weinberg mechanism, generating the mass
for the $U(1)_{B-L}$ gauge boson ($Z^\prime$ boson) and the right-handed
neutrinos. Through a small negative coupling between the SM Higgs doublet and
the $B-L$ Higgs field, the negative mass term for the SM Higgs doublet is
generated and the electroweak symmetry is broken. In this model context, we
investigate the electroweak vacuum instability problem in the SM. It is known
that in the classically conformal U(1)$_{B-L}$ extension of the SM, the
electroweak vacuum remains unstable in the renormalization group analysis at
the one-loop level. In this paper, we extend the analysis to the two-loop
level, and perform parameter scans. We identify a parameter region which not
only solve the vacuum instability problem, but also satisfy the recent ATLAS
and CMS bounds from search for $Z^\prime$ boson resonance at the LHC Run-2.
Considering self-energy corrections to the SM Higgs doublet through the
right-handed neutrinos and the $Z^\prime$ boson, we derive the naturalness
bound on the model parameters to realize the electroweak scale without
fine-tunings.
