Acosta is a former US attorney in Miami who served on the National Labor Relations Board in the Bush administration and headed the civil rights section in the Department of Justice.
"Mr. Acosta's nomination is off to a good start because he's already been confirmed by the Senate three times", Alexander said.
The news of Puzder's withdrawal comes amid reports that a number of Republicans meant to vote against him.
Democrats had come out strongly against Puzder, as they had with other contentious Trump nominees Jeff Sessions and Betsy DeVos.
For an administration that celebrates incompetence and Russian toadies for its top picks, the surprise announcement of Trump's new Labor pick demonstrates a shocking level of ability.
Puzder withdrew from consideration after Politico revealed a tape of his ex-wife, Lisa Fierstein, on Oprah Winfrey's show in 1990 in which she discussed Puzder assaulting her and vowing revenge on her for going public.
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.
He also served on the National Labor Relations Board for a year - from 2002 to 2003 - as a Republican member of the board under President George W. Bush.
Puzder was far from assured confirmation in the Senate.
Acosta, if confirmed, would be the first Hispanic in Trump's Cabinet.
Additional tactics include reports that individuals have been paid to protest Puzder; many at recent protests are even unaware of who Andy Puzder even is.
Senate majority leader Mitch McConnell informed the White House on Wednesday that he couldn't garner enough Republican votes to secure his confirmation.
Puzder's opponents had long laid the groundwork for their case that workers at Puzder's fast food business, including Hardee's and Carl's Jr., were treated shabbily.