Acosta, a lawyer from Florida, is the first Latino in Trump's cabinet.
The CKE Restaurants chief executive has also been criticised for his remarks on women and employees at his employees.
Acosta, the dean of the law school at Florida International University, served as an assistant attorney general for the Civil Rights Division under President George W. Bush and is a former US attorney for the Southern District of Florida.
According to a pool report, he will use the occasion to unveil his new pick for secretary of Labor.
Puzder said "while I won't be serving in the administration, I fully support the President and his highly qualified team". He has also served on the National Labor Relations Board and led the Justice Department's civil rights division.
Puzder confirmed his withdrawal on Twitter.
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.
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. Of course, even in that instance, Puzder saw immigrants as people who were just happy to have a job, according to comments he made to an audience at an American Enterprise Institute breakfast.
President Donald Trump's nominee for labour secretary Andrew Puzder, has withdrawn from consideration on the eve of his confirmation hearing. CBS News reported earlier Wednesday that a person close to Puzder said he was "very exhausted of the abuse".
His confirmation hearing was scheduled to take place on Thursday, but many Republicans had raised concerns about Puzder's failure to pay taxes on a former housekeeper who was not authorized to work in the United States.
Puzder married his current wife, Deanna Descher, in 1987.