We consider the minimal U(1)' extension of the standard model (SM) with the classically conformal invariance, where an anomaly-free U(1)' gauge symmetry is introduced along with three generations of right-handed neutrinos and a U(1)' Higgs field. Since the classically conformal symmetry forbids all dimensional parameters in the model, the U(1)' gauge symmetry is broken by the Coleman-Weinberg mechanism, generating the mass terms of the U(1)' gauge boson (Z' boson) and the right-handed neutrinos. Through a mixing quartic coupling between the U(1)' Higgs field and the SM Higgs doublet field, the radiative U(1)' gauge symmetry breaking also triggers the breaking of the electroweak symmetry. In this model context, we first investigate the electroweak vacuum instability problem in the SM. Employing the renormalization group equations at the two-loop level and the central values for the world average masses of the top quark (m(t) = 173.34 GeV) and the Higgs boson (m(h) = 125.09 GeV), we perform parameter scans to identify the parameter region for resolving the electroweak vacuum instability problem. Next we interpret the recent ATLAS and CMS search limits at the LHC Run-2 for the sequential Z' boson to constrain the parameter region in our model. Combining the constraints from the electroweak vacuum stability and the LHC Run-2 results, we find a bound on the Z' boson mass as m(Z') greater than or similar to 3.5 TeV. We also calculate self-energy corrections to the SM Higgs doublet field through the heavy states, the right-handed neutrinos and the Z' boson, and find the naturalness bound as m(Z') less than or similar to 7 TeV, in order to reproduce the right electroweak scale for the fine-tuning level better than 10%. The resultant mass range of 3.5 TeV less than or similar to m(Z') less than or similar to 7 TeV will be explored at the LHC Run-2 in the near future.