Old English weddian "to pledge oneself, covenant to do something, vow; betroth, marry," also "unite (two other people) in a marriage, conduct the marriage ceremony," from Proto-Germanic *wadja (source also of Old Norse veðja, Danish vedde "to bet, wager," Old Frisian weddia "to promise," Gothic ga-wadjon "to betroth"), from PIE root *wadh- (1) "to pledge, to redeem a pledge" (source also of Latin vas, genitive vadis "bail, security," Lithuanian vaduoti "to redeem a pledge"), which is of uncertain origin.
The sense has remained closer to "pledge" in other Germanic languages (such as German Wette "a bet, wager"); development to "marry" is unique to English. "Originally 'make a woman one's wife by giving a pledge or earnest money', then used of either party" [Buck]. Passively, of two people, "to be joined as husband and wife," from c. 1200. Related: Wedded; wedding.