The Universe is filled with a diffuse background of gamma-ray radiation, the origin of which remains one of the unsolved puzzles of cosmology. Less than one-quarter of the gamma-ray flux can be attributed to unresolved discrete sources, such as active galactic nuclei; the remainder appears to constitute a truly diffuse background. Here we show that the shock waves induced by gravity in the gas of the intergalactic medium, during the formation of large-scale structures like filaments and sheets of galaxies, produce a population of highly relativistic electrons. These electrons scatter a small fraction of the cosmic microwave background photons in the local Universe up to gamma-ray energies, thereby providing the gamma-ray background. The predicted diffuse flux agrees with the observed background across more than four orders of magnitude in photon energy, and the model predicts that the gamma-ray background, though generated locally, is isotropic to better than five per cent on angular scales larger than a degree. Moreover, the agreement between the predicted and observed background fluxes implies a mean cosmological density of baryons that is consistent with Big Bang nucleosynthesis.