By performing large-scale molecular-dynamics simulations of clusters of MgO we investigate the fundamental physics of melting, and the effects of pressure on melt and crystal structure along the melting curve from zero pressure to 300 GPa. We find that melting occurs at constant root-mean-squared (rms) displacements relative to the average near-neighbor distance over the entire pressure range, in agreement with Lindemann's predictions of 1910, and contrary to previous studies that indicate failure of Lindemann's law for nonmonatomic substances. The high-pressure validity of Lindemann's law supports one-phase models for melting. The liquid structure varies along the melting curve, becoming more crystal-like with increasing pressure with average coordinate number changing from 4.5 to 6, and \ensuremath{\Delta}V/${\mathit{V}}_{\mathit{c}}$ tends to zero with increasing pressure. Trends in thermodynamic functions and structure indicate that in the extreme pressure limit, melting is characterized only by dynamical changes such as onset of rapid diffusion, and not by local structural changes, since high pressure favors efficient packing of the liquid as well as the solid.