Christine Owens, executive director of the non-profit National Employment Law Project, said Acosta "deserves a thorough vetting" on labor and employee issues.
If confirmed, Acosta would be the first Hispanic member of Mr Trump's Cabinet.
Politico reported Puzder announced his withdrawal after Republican senators told White House officials Puzder's nomination would not make it through the Senate.
The only confirmation question I would have is why wasn't the confirmable Acosta nominated in the first place, instead of Puzder?
"The man I'll be announcing for labor is a star, great person", Trump said as he opened a meeting with some of his staunchest supporters in the House. Should Acosta be confirmed by the Senate, his appointment would remedy that complaint.
Puzder, who lives in Franklin, Tenn., is CEO of CKE Restaurants, the parent company of the Hardee's and Carl's Jr. fast-food chains. They included five members of the Senate Health, Education, Labor and Pensions (HELP) committee, which was scheduled to hold a confirmation hearing for Puzder on February 16.
TVNZ's US Correspondent says Trump is going back on the campaign trail for a 2020 re-election.
Puzder's critics have said labor practices at his restaurants and his opposition to raising the minimum wage and expanding overtime eligibility should disqualify him from leading the Department of Labor. He also is dean of Florida International University's School of Law, a position he has held since 2009, according to his biography on the school's website.
Compounding the domestic abuse allegation, Puzder was accused of sexism.