Goodman, 33, resigned for "inappropriate conduct" shortly after being caught earlier this week.
His Twitter account has now been locked and his campaign website has been taken down.
On Twitter, he previously described himself as "Christian. American. Conservative. Republican. Husband to @Beth1027" and often tweeted about how "natural marriage" was only between a man and a woman.
On his campaign website, he outlined his views on "family": "Healthy, vibrant, thriving, values-driven families are the source of Ohio's proud history and the key to Ohio's future greatness. The ideals of a loving father and mother, a committed natural marriage, and a caring community are well worth pursuing and protecting."