  Star-forming galaxies have been predicted to contribute considerably to the
diffuse gamma-ray background as they are guaranteed reservoirs of cosmic rays.
Assuming that the hadronic interactions responsible for high-energy gamma rays
also produce high-energy neutrinos and that O(100) PeV cosmic rays can be
produced and confined in starburst galaxies, we here discuss the possibility
that star-forming galaxies are also the main sources of the high-energy
neutrinos observed by the IceCube experiment. First, we compute the diffuse
gamma-ray background from star-forming galaxies, adopting the latest Herschel
PEP/HerMES luminosity function and relying on the correlation between the
gamma-ray and infrared luminosities reported by Fermi observations. Then we
derive the expected intensity of the diffuse high-energy neutrinos from
star-forming galaxies including normal and starburst galaxies. Our results
indicate that starbursts, including those with active galactic nuclei and
galaxy mergers, could be the main sources of the high-energy neutrinos observed
by the IceCube experiment. We find that assuming a cosmic-ray spectral index of
2.1-2.2 for all starburst-like galaxies, our predictions can be consistent with
both the Fermi and IceCube data, but larger indices readily fail to explain the
observed diffuse neutrino flux. Taking the starburst high-energy spectral index
as free parameter, and extrapolating from GeV to PeV energies, we find that the
spectra harder than E^(-2.15) are likely to be excluded by the IceCube data,
which can be more constraining than the Fermi data for this population.
