The withdrawal signifies the first casualty of the Trump administration's Cabinet picks, as the Senate was expected to confirm Puzder later in the week.
Puzder is CEO of the CKE restaurant chain, which includes Hardee's and Carl Jr's, and the idea of a fast food boss protecting workers' rights had raised eyebrows from the start.
After five separate delays, his confirmation hearing before the Senate Health, Education, Labor and Pensions Committee was scheduled for Thursday.
The Independent notes that he would have needed 50 out of the 52 Republicans in the Senate to vote for him, but some Republicans do not stand on his side.
Puzder has been dogged by previous accusations of domestic assault. How likely is it that Acosta will be confirmed as Labor secretary?
The nominee has also faced concerns about his views on overtime and the minimum wage.
The president is also said to have offered the position of National Security Adviser, which fell vacant on Michael Flynn's ouster, to Vice-Admiral Robert Harward, who according to some reports, has asked for time to consider.
Acosta's nomination came a day after fast-food CEO Andrew Puzder withdrew after it became clear he had insufficient Republican support.
Decades ago, Puzder's ex-wife, Lisa Fierstein, brought abuse charges against him, which she later dropped.
"I urge President Trump to select a new Secretary of Labor nominee who will work with me and my fellow attorneys general to ensure workers receive the wages they are owed and stay safe on the job".
Puzder's hearing had been postponed for some weeks as he had taken so long to provide the required paperwork. He also admitted to employing an undocumented woman as a housekeeper. In the 1990 episode, wearing an elaborate disguise, Fierstein detailed physical abuse she said Puzder inflicted on her during their marriage. When he heard the news of Puzder' withdrawal, Schumer was quick to react, saying it is a "victory for the American worker".