  We extend the Zee model by introducing a vector-like lepton doublet and a
flavor dependent global $U(1)$ symmetry. Flavor changing neutral currents in
the quark sector can be naturally forbidden at tree level due to the $U(1)$
symmetry, while sufficient amount of lepton flavor violation is provided to
explain current neutrino oscillation data. In our model, additional sources of
CP-violation appear in the lepton sector, but their contribution to electric
dipole moments is much smaller than current experimental bounds due to the
Yukawa structure constrained by the $U(1)$ symmetry. We find that there is a
parameter region where the strongly first order electroweak phase transition
can be realized, which is necessary for the successful scenario of the
electroweak baryogenesis in addition to new CP-violating phases. In the
benchmark points satisfying neutrino data, lepton flavor violation data and the
strongly first order phase transition, we show that an additional CP-even Higgs
boson $H$ mainly decays into a lighter CP-odd Higgs boson $A$, i.e., $H \to AZ$
or $H \to AA$ with a characteristic pattern of lepton flavor violating decays
of $A$.
