Emily Moreno had previously accused her ex-husband, an Ohio congressman, of domestic abuse, including throwing hot water on her and holding a gun to her head. He denies the allegations.
The Trump administration is seeking removal of a woman accused of helping family members who sought to carry out a mass shooting inspired by the Islamic State.
Barnes withdrew from the race as the Milwaukee Journal Sentinel was preparing to report that the state Democratic Party last year reviewed anonymous allegations about his sexual encounters.
Two Republicans, who are set to leave the Senate next year, are insisting that the president’s IRS deal must be rewritten before they will vote for the nominee.