Following our former study of the $\mathrm{C}\mathrm{O}∕\mathrm{Ni}\mathrm{O}(100)$ system [Phys. Rev. B 69, 075413 (2004)], we have investigated the geometrical, electronic, and magnetic properties of NO adsorbed on NiO(100) to further elucidate the role of strong electron correlations in the substrate on the chemisorption of small molecules. As for $\mathrm{C}\mathrm{O}∕\mathrm{Ni}\mathrm{O}(100)$, density-functional theory (DFT, both in the local-density and in the generalized gradient approximations) strongly overestimates the adsorption strength, but in contrast to CO, the strongly tilted adsorption geometry of NO is correctly predicted even at the DFT level. If strong electronic correlation in NiO are described by a $\mathrm{DFT}+U$ approach, the shift of the occupied Ni $d$ states to higher binding energies nearly completely suppresses donation from the occupied $5\ensuremath{\sigma}$ molecular orbital to the Ni ${d}_{{z}^{2}}$ states and backdonation from the Ni ${d}_{xy,xz}$ orbitals to the partially filled (NO) $2\ensuremath{\pi}*$ molecular orbital, the conventional bonding mechanism between a $p$-bonded molecular dimer and a transition-metal atom. $\mathrm{DFT}+U$ calculations predict NO to be almost unbound in an upright configuration. In a tilted configuration, a hybridization of the $2\ensuremath{\pi}*$ molecular orbital with the Ni ${d}_{{z}^{2}}$ states, which is symmetry forbidden in the upright configuration, provides an alternative mechanism for the formation of a weak covalent adsorbate-substrate bond. The energetic, geometric, vibrational, and magnetic properties of the adsorbate/substrate complex are strongly influenced by this alternative mechanism. Altogether our results demonstrate that, as suggested by earlier studies, the on-site Coulomb repulsion in the Ni $d$ band plays a decisive role in the description of adsorption on transition-metal oxides and that $\mathrm{DFT}+U$ provides a sound description of chemisorption on these difficult systems.