He praised Acosta for his "tremendous career."
"I've wished him the best," Trump said. "I think he'll be a tremendous secretary of labor."
Acosta, a protege of conservative Supreme Court Justice Samuel Alito, was appointed to the National Labor Relations Board by President George W. Bush. He also served as assistant attorney general for the Justice Department's civil rights division under Bush's administration.
If Acosta is confirmed, he will be the first Latino member of Trump's cabinet.
Andrew Puzder withdrew his nomination for secretary of labor on Wednesday amid uncertain support from Republican senators.