  We extend the Zee model, where tiny neutrino masses are generated at the one
loop level, to a supersymmetric model with R-parity conservation. It is found
that the neutrino mass matrix can be consistent with the neutrino oscillation
data thanks to the nonholomorphic Yukawa interaction generated via one-loop
diagrams of sleptons. We find a parameter set of the model, where in addition
to the neutrino oscillation data, experimental constraints from the lepton
flavor violating decays of charged leptons and current LHC data are also
satisfied. In the parameter set, an additional CP-even neutral Higgs boson
other than the standard-model-like one, a CP-odd neutral Higgs boson, and two
charged scalar bosons are light enough to be produced at the LHC and future
lepton colliders. If the lightest charged scalar bosons are mainly composed of
the SU(2)_L-singlet scalar boson in the model, they would decay into e nu and
mu nu with 50% of a branching ratio for each. In such a case, the relation
among the masses of the charged scalar bosons and the CP-odd Higgs in the
minimal supersymmetric standard model approximately holds with a radiative
correction. Our model can be tested by measuring the specific decay patterns of
charged scalar bosons and the discriminative mass spectrum of additional scalar
bosons.
