One of the posts on businessman Mike Parry's account called Obama a "Power Hungry Arrogant Black Man," while the other said "whats with the Dems and Pedophiles?" Both were dated in May.
A statement issued Wednesday by Parry says he's taking responsibility for "all mistakes in my past tweets."
He says he apologizes for posts "written in haste and out of the frustration I felt for the out of control spending in Washington."
Parry is running in a Jan. 26 special election in southern Minnesota. The winner will finish the term of longtime Republican state Sen. Dick Day, who resigned to become a lobbyist.