By performing simulations of clusters of MgO using a non‐empirical potential we investigate the fundamental physics of melting, the effects of pressure, melt and crystal structure, and themodynamic properties along the melting curve from zero pressure to 300 GPa. We find the surprising result that melting occurs when r.m.s. displacements of the ions reach about 18% of the near‐neighbor distance over the entire pressure range, in agreement with Lindemann’s predictions of 1910, contrary to previous studies which indicate failure of Lindemann’s law for non‐monatomic substances. The present results indicate clearly that melting is related to intrinsic instability of crystals similar to lattice instabilities that lead to solid‐solid phase transitions.