Old English gretan "to come in contact with" in any sense ("attack, accost" as well as "salute, welcome," and "touch, take hold of, handle," as in hearpan gretan "to play the harp"), "seek out, approach," from West Germanic *grotjan (source also of Old Saxon grotian, Old Frisian greta, Dutch groeten, Old High German gruozen, German grüßen "to salute, greet"), of uncertain origin.
In English, German, and Dutch, the primary sense has become "to salute," but the word once had much broader meaning. Perhaps originally "to resound" (via notion of "cause to speak"), causative of Proto-Germanic *grætanan, root of Old English grætan (Anglian gretan) "weep, bewail," from PIE *gher- (2) "to call out." Greet still can mean "cry, weep" in Scottish & northern England dialect, though this might be from a different root. Grætan probably also is the source of the second element in regret. Related: Greeted; greeting.