We derive the contribution to the extragalactic gamma-ray background (EGB) from AGN winds and star-forming galaxies by including a physical model for the gamma-ray emission produced by relativistic protons accelerated by AGN-driven and supernova-driven shocks into a state-of-the-art semi-analytic model of galaxy formation. This is based on galaxy interactions as triggers of AGN accretion and starburst activity and on expanding blast wave as the mechanism to communicate outwards the energy injected into the interstellar medium by the active nucleus. We compare the model predictions with the latest measurement of the EGB spectrum performed by the Fermi-LAT in the range between 100 MeV and 820 GeV. We find that AGN winds can provide ~35$pm$15% of the observed EGB in the energy interval E_{gamma}=0.1-1 GeV, for ~73$pm$15% at E_{gamma}=1-10 GeV, and for ~60$pm$20% at E_{gamma}>10 GeV. The AGN wind contribution to the EGB is predicted to be larger by a factor of 3-5 than that provided by star-forming galaxies (quiescent plus starburst) in the hierarchical clustering scenario. The cumulative gamma-ray emission from AGN winds and blazars can account for the amplitude and spectral shape of the EGB, assuming the standard acceleration theory, and AGN wind parameters that agree with observations. We also compare the model prediction for the cumulative neutrino background from AGN winds with the most recent IceCube data. We find that for AGN winds with accelerated proton spectral index p=2.2-2.3, and taking into account internal absorption of gamma-rays, the Fermi-LAT and IceCube data could be reproduced simultaneously.
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