Trump tweeted the news of Moore's withdrawal, only hours after Moore had told two news organizations that he was still seeking the board seat and still had the White House's support.
The president announced otherwise Thursday afternoon on Twitter.
"Steve won the battle of ideas including Tax Cuts and deregulation which have produced non-inflationary prosperity for all Americans," Trump said. "I've asked Steve to work with me toward future economic growth in our Country."
In a note to Trump that he released later, Moore said the "unrelenting attacks on my character have become untenable for me and my family and three more months of this would be too hard on us."
"I am always at your disposal," he concluded.
Speaking later to Fox Business Network, Moore offered a combative defense of his candidacy for the Fed. "It was very disappointing that this couldn't go forward," he said.
Numerous Republican senators had said they objected to Moore's disparaging past writings about women or had sidestepped questions about whether they would back him. In recent weeks, Moore said he regretted the writings and said they had been meant as humor columns.
The Senate's second-ranking Republican, Sen. John Thune of South Dakota, had said Wednesday that Moore "has issues" in the Senate, which must confirm any nominees for the Fed's board.
Sen. Shelley Moore Capito, R-W.Va., said it was "hard to look past" Moore's previous statements, while Sen. Marsha Blackburn, R-Tenn., said his comments were a topic on which she would have questioned him.
In 2000, in a column for the Washington Times, Moore wondered why women "showed up in droves in tight skirts" at college parties if "they were so oppressed and offended by drunken, lustful frat boys."
He also said women should not cover basketball games on television unless they wore revealing clothes.
Yet finding someone who fits the mold Trump seems to be seeking in a Fed governor is problematic. The president appears to be aiming for a reliable political ally who will push the Fed to cut short-term interest rates. Most traditional right-leaning economists, though, have pushed for higher interest rates for most of the past decade. Any, like Moore, who have reversed themselves to embrace the dovish approach favored by Trump might have a hard time winning Senate support.
Trump had also named Herman Cain, a former presidential candidate and business executive, for a second open seat on the Fed's board. But Cain withdrew last week after coming under renewed scrutiny for allegations of sexual harassment and infidelity which first surfaced during the campaign.
Some experts say that David Beckworth, a research fellow at George Mason University's Mercatus Center, might fit what Trump is seeking — a right-of-center economist who has pushed for low rates for most of the past decade.
Beckworth has declined to comment.
Kathleen Hall Jamieson, director of the University of Pennsylvania's Annenberg Public Policy Center, suggested that Moore's defeat reflected the proper functioning of the political system and the key role of the news media in illuminating Moore's written commentaries.
"This is an example of the system working as it should," Jamieson said in an interview. "Republicans spoke up."
As an adviser to Trump's presidential campaign, Moore helped design the 2017 tax cuts. Yet his candidacy immediately met widespread skepticism about whether he was qualified for a Fed board position and concerns about his background as a highly politicized commentator on economic issues.
Sung Won Sohn, a finance and economics professor at Loyola Marymount University in Los Angeles, said Thursday that he thought Moore's withdrawal was a positive development for the Fed.
"The most important asset a central bank has is its reputation and credibility," Sohn said. He contended that putting Moore on the Fed board would have imperiled those standards.
After Trump named Moore as a potential nominee in late March, Gregory Mankiw, a Harvard professor who was a top adviser to President George W. Bush, observed that Moore "does not have the intellectual gravitas for this important job" and urged the Senate to keep him off the Fed.
Moore has argued that the Fed should follow changes in the prices of commodities, such as oil or farm goods, and raise rates if those prices rose, to stem inflation. Yet that approach would have caused the Fed to raise rates in 2008 as oil prices spiked, which would have worsened the Great Recession.
AP Business Writers Martin Crutsinger and Marcy Gordon contributed to this report.