My dad and stepmom married in Greece, then moved to Arizona. My dad is a U.S. citizen, however my stepmom is not. They never registered their marriage in Arizona, and did not apply for my stepmom to become a U.S. Citizen. My stepmom filed for a divorce, and now lives in Texas. Would they still have to get a divorce as they never did register their marriage in the U.S. or Arizona?

yes they need a divorce.