Sen. John McCain, R-Ariz., called the allegations against Moore "disqualifying."
"He should immediately step aside and allow the people of Alabama to elect a candidate they can be proud of," McCain said on Twitter.
It is too late for Moore to legally withdraw from the Senate race. The ballots are already printed, and many absentee voters have already submitted theirs.
Republicans were scrambling to figure out whether Sen. Luther Strange, who now holds the seat, could run a viable campaign as a write-in candidate against Moore's Democratic rival, Doug Jones.
Strange, who was appointed to the seat to replace Sen. Jeff Sessions when President Donald Trump named Sessions U.S. attorney general, lost the Republican primary to Moore.
On Capitol Hill, Strange told reporters that the Post report was "very, very disturbing." He ignored a question on whether he might launch a write-in campaign.
Three other women told the Post that Moore, when he was in his early 30s, pursued them when they were between 16 and 18 years old, but never forced sex on them. The age of consent for sex under Alabama law is 16, as it was at the time of the alleged incidents.
None of the women came forward to the Post. A reporter for the newspaper heard allegations that Moore had sought relationships with teenage girls, and the Post subsequently found and interviewed the four women.
Moore told the Post in a written statement: "These allegations are completely false and are a desperate political attack by the National Democrat Party and The Washington Post on this campaign."
Moore has run as a firebrand stressing cultural appeals. He lamented "the awful calamity of abortion and sodomy and perverse behavior and murders and shootings and road rage" as "a punishment inflicted upon us for our presumptuous sins."
He has also referred to Native Americans and Asian Americans as "reds and yellows."
Moore was ousted from the Alabama Supreme Court in 2003 for defying a federal order to remove a Ten Commandments monument from a state courthouse. Voters elected him chief justice again in 2012, but he was suspended for refusing to follow the U.S. Supreme Court ruling that legalized same-sex marriage. He later resigned.
In Alabama, supporters came to Moore's defense. State Auditor Jim Zeigler invoked the Bible when he told The Washington Examiner: "Take Joseph and Mary. Mary was a teenager and Joseph was an adult carpenter. They became parents of Jesus."
"There's just nothing immoral or illegal here," he said. "Maybe just a little bit unusual."
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