Acosta, the dean of Florida International University Law School, is a Harvard-trained lawyer who has served as assistant attorney general for the Justice Department's Civil Rights Division and as US attorney for the Southern District of Florida. He has served on the National Labor Relations Board and as a federal prosecutor in Florida.
CBS News Chief White House Correspondent Major Garret said that a source very close to President Donald Trump's Labor Department nominee told him that he expects Puzder to withdraw because, "he's very exhausted of the abuse".
"After careful consideration and discussions with my family, I am withdrawing my nomination for secretary of Labor", Puzder said in a statement emailed by his spokesman. Multiple Republican senators also pulled their support. Acosta is the first Hispanic that Trump has chosen for his cabinet.
In an 18 January letter to the Senate committee that had been due to hold Mr Puzder's confirmation hearing, Ms Fierstein said: "Andy is not and was not abusive or violent".
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Puzder's nomination had come under scrutiny in recent weeks, first after revelations that he had employed an undocumented worker and more recently a tape provided to senators by the Oprah Winfrey Network that accused him of physical abuse.
Acosta, the only son of Cuban immigrants, gradually reconnected with Miami and cultivated strong relationships in the legal and law enforcement community to build up the US attorney's office in his image.
Acosta, a native of Miami who graduated from Harvard Law School, became dean of the FIU law school in 2009, according to his bio on the college's website.
Like Trump's immigration ban, opposing Puzder's nomination was not a political issue, but a right/wrong issue. She has since retracted the allegations, which Puzder has always denied. In the 1990 episode, wearing an elaborate disguise, Fierstein detailed physical abuse she said Puzder inflicted on her during their marriage.