1540s, "influential, respected; marked by weighty dignity," from Middle French grave (Old French greve "terrible, dreadful," 14c.), from Latin gravis, "heavy, ponderous, burdensome, loaded; pregnant;" of matters, "weighty, important;" of sounds, "deep, low, bass;" figuratively "oppressive, hard to bear, troublesome, grievous," from PIE root *gwere- (1) "heavy."
In English, the sense "solemn, sober" is from 1580s; of immaterial things, "important, serious" 1590s. Greek barys (opposed to kouphos) also was used figuratively, of suffering, sorrow, sobbing, and could mean "oppressive, burdensome, grave, dignified, impressive." The noun meaning "accent mark over a vowel" is c. 1600, from French.