  We suggest the so-called bosonic seesaw mechanism in the context of a
classically conformal $U(1)_{B-L}$ extension of the Standard Model with two
Higgs doublet fields. The $U(1)_{B-L}$ symmetry is radiatively broken via the
Coleman-Weinberg mechanism, which also generates the mass terms for the two
Higgs doublets through quartic Higgs couplings. Their masses are all positive
but, nevertheless, the electroweak symmetry breaking is realized by the bosonic
seesaw mechanism. We analyze the renormalization group evolutions for all model
couplings, and find that a large hierarchy among the quartic Higgs couplings,
which is crucial for the bosonic seesaw mechanism to work, is dramatically
reduced toward high energies. Therefore, the bosonic seesaw is naturally
realized with only a mild hierarchy, if some fundamental theory, which provides
the origin of the classically conformal invariance, completes our model at some
high energy, for example, the Planck scale. The requirements for the
perturbativity of the running couplings and the electroweak vacuum stability in
the renormalization group analysis as well as for the naturalness of the
electroweak scale, we have identified the regions of model parameters. For
example, the scale of the $U(1)_{B-L}$ gauge symmetry breaking is constrained
to be $\lesssim 100$ TeV, which corresponds to the extra heavy Higgs boson
masses to be $\lesssim 2$ TeV. Such heavy Higgs bosons can be tested at the
Large Hadron Collider in the near future.
