  We investigate numerically the contribution to the cosmic gamma-ray
background from cosmic-rays ions and electrons accelerated at intergalactic
shocks associated with cosmological structure formation. We show that the
kinetic energy of accretion flows in the low-red-shift inter-galactic medium is
thermalized primarily through moderately strong shocks, which allow for an
efficient conversion of shock ram pressure into cosmic-ray pressure.
Cosmic-rays accelerated at these shocks produce a diffuse gamma-ray flux which
is dominated by inverse Compton emission from electrons scattering off cosmic
microwave background photons. Decay of neutral pi-mesons generated in p-p
inelastic collisions of the ionic cosmic-ray component with the thermal gas
contribute about 30% of the computed emission. Based on experimental upper
limits on the photon flux above 100 MeV from nearby clusters we constrain the
efficiency of conversion of shock ram pressure into relativistic CR electrons
to < 1%. Thus, we find that cosmic-rays of cosmological origin can generate an
overall significant fraction of order 20% and no more than 30% of the measured
gamma-ray background.
