Analysis of Hadean and Archean rocks for O-16-O-17-O-18 isotopes demonstrates that the Terrestrial Mass Fractionation Line of oxygen isotopes has had the same slope and intercept for at least the past 4.0 and probably for as long as 4.2Ga. The homogenization of oxygen isotopes required to produce such long-lived consistency was most easily established by mixing in a terrestrial magma ocean. The measured identical oxygen isotope mass fractionation lines for Earth and Moon suggest that oxygen isotope reservoirs of both bodies were homogenized at the same time during a giant moon-forming impact. But other sources of heat for global melting cannot be excluded such as bolide impacts during early accretion of proto-Earth, the decay of short-lived radioactive isotopes, or the energy released during segregation of core from mantle.