late 15c., "act of avenging, revenge," from Old French vindicacion "vengeance, revenge" and directly from Latin vindicationem (nominative vindicatio) "act of claiming or avenging," noun of action from past participle stem of vindicare "lay claim to, assert; claim for freedom, set free; protect, defend; avenge" (related to vindicta "revenge"), probably from vim dicare "to show authority," from vim, accusative of vis "force" (see vim) + dicare "to proclaim" (from PIE root *deik- "to show," also "pronounce solemnly," and see diction). Meaning "justification by proof, defense against censure" is attested from 1640s.

Latin, "(with) God (as our) defender," national motto of the Confederate States of America, from ablative of Deus "God" (see Zeus) + ablative of present participle of  vindicare "to liberate; to act as avenger; protect, defend" (see vindication).

late 14c., revengen, "avenge oneself," from Old French revengier, revenger, variants of revenchier "take revenge, avenge" (13c., Modern French revancher), from re-, here perhaps an intensive prefix (see re-), + vengier "take revenge," from Latin vindicare "to lay claim to, avenge, punish" (see vindication). Transitive sense of "take vengeance on account of" is from early 15c. Related: Revenged; revenging; revengement.

To avenge is "to get revenge" or "to take vengeance"; it suggests the administration of just punishment for a criminal or immoral act. Revenge seems to stress the idea of retaliation a bit more strongly and implies real hatred as its motivation. ["The Columbia Guide to Standard American English," 1993]

"the act of exonerating from a charge of fault or crime; vindication," 1715, noun of action from exculpate.