The merger of dark matter halos and the gaseous structures embedded in them, such as proto-galaxies, galaxies, and groups and clusters of galaxies, results in strong shocks that are capable of accelerating cosmic rays (CRs) to ∼ > 10 PeV. These shocks will produce high-energy neutrinos and γ-rays through inelastic pp collisions. In this work, we study the contributions of these halo mergers to the diffuse neutrino flux and to the non-blazar portion of the extragalactic γ-ray background. We formulate the redshift dependence of the shock velocity, galactic radius, halo gas content and galactic/intergalactic magnetic fields over the dark matter halo distribution up to a redshift z = 10. We find that high-redshift mergers contribute a significant amount of the cosmic-ray luminosity density, and the resulting neutrino spectra could explain a large part of the observed diffuse neutrino flux above 0.1 PeV up to ∼ PeV. We also show that our model can somewhat alleviate tensions with the extragalactic γ-ray background. First, since a larger fraction of the CR luminosity density comes from high redshifts, the accompanying γ-rays are more strongly suppressed through γγ annihilations with the cosmic microwave background (CMB) and the extragalactic background light (EBL). Second, mildly radiative-cooled shocks may lead to a harder CR spectrum with spectral indices of 1.5 ∼ < s ∼ < 2.0. Our study suggests that halo mergers, a fraction of which may also induce starbursts in the merged galaxies, can be promising neutrino emitters without violating the existing Fermi γ-ray constraints on the non-blazar component of the extragalactic γ-ray background.