  Ultrahigh energy cosmic ray (UHECR) protons interacting with the cosmic
microwave background (CMB) produce UHE electrons and gamma-rays that in turn
initiate electromagnetic cascades on CMB and infrared photons. As a result, a
background of diffuse isotropic gamma radiation is accumulated in the energy
range $E\lsim 100$ GeV. The Fermi-LAT collaboration has recently reported a
measurement of the extragalactic diffuse background finding it less intense and
softer than previously measured by EGRET. We show that this new result
constrains UHECR models and the flux of cosmogenic neutrinos. In particular, it
excludes models with cosmogenic neutrino fluxes detectable by existing neutrino
experiments, while next-generation detectors as e.g. JEM-EUSO can observe
neutrinos only for extreme parameters.
