Trump said Acosta, who did not appear with the president, "has had a tremendous career." He noted that, unlike Puzder, Acosta has been confirmed by the Senate three times and "did very, very well."
If confirmed anew by the Senate, Acosta would become the first Hispanic member of Trump's Cabinet. He is now dean of the Florida International University law school.
Fast-food CEO Andrew Puzder pulled out yesterday after it became clear he lacked the votes to win Senate confirmation.
"I wish him the best," Trump said. At a White House event earlier, he had described Acosta - without identifying him -- as a "star" and a "great person."
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Acosta has served on the National Labor Relations Board and as a federal prosecutor in Florida. He was named assistant attorney general for civil rights by President George W Bush.
Puzder pulled out of the process on the eve of his confirmation hearing, which had been scheduled for Thursday, because Republicans balked at an array of personal and professional issues that dogged him. Puzder said he had employed -- and belatedly paid taxes on -- a housekeeper not authorized to work in the United States.
Contentious confirmation fights, a botched rollout of Trump's refugee order and the ouster this week of National Security Adviser Michael Flynn have nearly paralyzed the administration.
With regard to Puzder, Republicans grumbled about the stream of "distractions," including the torrent of criticism about his personal life and record as CEO of CKE Restaurants, Inc.
What ultimately drained Puzder of enough Republican support for confirmation was his acknowledgment -- well after Trump had become president -- of employing a housekeeper who was not authorized to work in the US.
Puzder said he paid the taxes as soon as he found they were owed, but there was no explanation of why he didn't know or pay for five years.
Spokesman George Thompson said yesterday that Puzder did not tell the White House about the housekeeper issue until after he had been nominated.
It's unclear whether Trump's aides asked the immigration question before the nomination even though such issues have sunk past presidential nominees. Trump has also taken a hard line on people in the US illegally.