Washington Republicans moved quickly Thursday to try to oust their candidate in the Alabama Senate race, former state Chief Justice Roy Moore, after a newspaper reported he had enticed an underage girl into a sexual encounter decades ago.

The Washington Post reported that a woman says Mr. Moore initiated the encounter in 1979, when she was 14 and he was 32. Three other women told the newspaper Mr. Moore pursued them when they were 16 to 18 years of age.

That was the last straw for Republicans who’d already been at odds with Mr. Moore, their nominee, on a host of other issues.

“The allegations against Roy Moore are deeply disturbing and disqualifying. He should immediately step aside and allow the people of Alabama to elect a candidate they can be proud of,” said Sen. John McCain, Arizona Republican.

Others didn’t go that far but said if the allegations turn out to be true, Mr. Moore should step aside.

Sen. Cory Gardner, chairman of the National Republican Senatorial Committee called the allegations “deeply troubling” and he and Senate Majority Leader Mitch McConnell both said Mr. Moore should drop out if there’s truth to the claims.

“If that’s true, he wouldn’t belong in the Senate,” said Sen. Richard Shelby, Alabama Republican.

Mr. Moore gave no indication he would accede to the demands and struck back at The Post on Twitter, saying they were his opponents in “a spiritual battle.”

“The forces of evil will lie, cheat, steal — even inflict physical harm — if they believe it will silence and shut up Christian conservatives like you and me,” Mr. Moore told supporters, urging them to stay behind him.

“Our children and grandchildren’s futures are on the line. So rest assured — I will NEVER GIVE UP the fight!” he tweeted.

Mr. Moore’s campaign said if the allegations were true, they would have emerged in previous campaigns for governor or chief justice.

Mr. Moore is running for the seat left vacant earlier this year when Jeff Sessions was tapped to be attorney general. Mr. Moore was the top vote-getter in the GOP primary and then won a runoff with Sen. Luther Strange, the man appointed to fill the seat between Mr. Sessions’ departure and the special election slated for Dec. 12.

The national GOP establishment — including President Trump — had fought Mr. Moore’s nomination, backing Mr. Strange in the primary and runoff.

But some conservative activists had rallied to the former state chief judge, arguing he was the most Trump-like candidate in the race.

Stephen K. Bannon, Mr. Trump’s former chief strategist who’s now back at his post at Breitbart.com, was among those backing Mr. Moore. On Thursday, Breitbart called out The Post for its story, saying that the paper had endorsed Mr. Moore’s opponent in the special election, Democratic candidate Doug Jones, a former U.S. attorney.

As chief judge, Mr. Moore gained national notoriety for defying orders that he remove a monument to the Ten Commandments from a judicial building.

Mr. Moore’s allies earlier this fall had said national party groups weren’t doing their part to help Mr. Moore get election, even after he became the GOP’s nominee.

That is unlikely to change now. The Senate Leadership Fund, a deep-pocketed political action committee with ties to Mr. McConnell, urged Alabama Republicans to find a way to remove Mr. Moore from the ballot.

“There is no place in our party for sexual predators,” said SLF President Steven Law.

• Tom Howell Jr. contributed to this article.

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