WASHINGTON — President Trump announced Monday that Herman Cain, one of his two embattled picks for the Federal Reserve Board, had withdrawn his name from consideration, even as his second candidate came under new scrutiny over his attitudes toward women.

Mr. Cain, a former pizza chain executive, made his decision as he battled old accusations of sexual harassment that had halted his 2012 presidential campaign.

His withdrawal bows to political reality in a moment when Mr. Trump has faced mounting criticism for tapping loyalists to join the historically independent Fed. And it moved a spotlight to the other man Mr. Trump has said he wants to put on the Fed, his economic adviser Stephen Moore, who faced new objections on Monday because of a series of magazine columns that denigrated women, including his wife at the time.

Mr. Moore has called the writings jokes, but the criticism suggests that questions of harassment and sexism could prove more consequential for Mr. Trump’s nominees than interest rates and other policy issues. In his columns, published in the early 2000s by the conservative magazine National Review, some of which were first reported by CNN, he complained that women are “sooo malleable” because his wife at the time voted for a Democrat, based on a campaign commercial.