1540s, "to cast off by divorce," also general, "reject, refuse to accept" (a person or thing), from Latin repudiatus, past participle of repudiare "to cast off, put away, divorce, reject, scorn, disdain," from repudium "divorce, rejection, a putting away, dissolution of marriage," from re- "back, away" (see re-) + pudium.
This is probably is related to pudere "cause shame to," a verb of unknown etymology. Barnhart, however, suggests it is related to pes/ped- "foot," in which case the original notion may be of kicking something away.
In reference to persons, "to disown," 1690s. Of opinions, conduct, etc., "to refuse to acknowledge, reject with condemnation," attested from 1824. Of debts by 1837. Earliest in English as an adjective meaning "divorced, rejected, condemned" (mid-15c.), from Latin repudiatus. Related: Repudiated; repudiating; repudiable.