The couple was arrested in Santa Barbara last September for allegedly causing more than $5,000 in damage to a home they once owned.
They did not appear at several court hearings and were later arrested in Vancouver, where they sought asylum. Quaid's wife, Evi Quaid, recently received her Canadian citizenship and sponsored his permanent residency.
Quaid and his wife claimed they were running away from star whackers - people they said hunt down celebrities.
Authorities in Santa Barbara had been seeking the return of the Quaids, but the state department has now decided not to pursue extradition.
Senior Deputy District Attorney Lee Carter said the "Independence Day" actor and his wife can be arrested if they return to the United States.
The Associated Press contributed to this story.