Even if Puzder did not withdraw, it is unlikely he'd win the votes to be confirmed.
According to CNN:
Top Senate Republicans have urged the White House to withdraw the Andrew Puzder nomination for labor secretary, a senior GOP source said, adding
there are four firm Republican no votes and possibly up to 12.
Puzder needs at least 50 votes to pass with the tie-breaking vote of Vice President Mike Pence, and Republicans only hold control of 52 seats.
Puzder, the CEO of the company that owns the Hardee's and Carl's Jr. fast food chains, has faced fierce opposition mostly from Democrats in part related to his position on labor issues as well as the fact that he employed an undocumented housekeeper.