WASHINGTON — Thousands of marchers converged in the nation’s capital and across the country on Saturday to show their support for abortion rights nearly two weeks after the leak of a Supreme Court draft opinion that would overturn Roe v. Wade.

A crowd of protesters gathered near the Washington Monument before marching to the Supreme Court, with some wearing shirts that read “Bans Off Our Bodies” and “Keep Abortion Safe and Legal.” They vowed to fight to preserve abortion rights, even as some accepted that Roe would most likely be overturned.

Colleen Lunsford, 42, a lawyer from Arlington, Va., brought her 5-year-old daughter, Orla. Pointing to her daughter, she said she attended the march for “her future and autonomy.”

“I’m terrified,” Ms. Lunsford said. “We did our best to elect a Democratic president and House and Senate, and this is still happening.”

More than 450 marches across the country were set to take place on Saturday, said Rachel O’Leary Carmona, the executive director of the Women’s March, a nonprofit organization that helped organize the event and other protests supporting women’s rights. Organizers had been planning a national march for abortion rights before the draft opinion leaked, but they fast-tracked the event after the opinion was published. Ms. O’Leary Carmona said she hoped the events would allow demonstrators to “build power, both civically and electorally.”

“Folks are mobilizing because they see that the hour is later than we thought,” she said.

The marches took place after the publication this month of the draft opinion, which showed that the Supreme Court appeared poised to overturn Roe, the landmark 1973 decision that guaranteed the right to abortion. The court’s ruling is not expected until June or early July.

With the midterm elections months away, President Biden and congressional Democrats are hoping to use the issue to energize voters. Democratic senators failed on Wednesday to advance legislation to guarantee abortion rights nationwide in the face of opposition from Republicans and one Democrat, Senator Joe Manchin III of West Virginia.

In Washington, Elizabeth Moser, 34, a communications specialist from Burke, Va., said she hoped the marches would galvanize voters and politicians.

Although she had been planning to vote in the midterms, she said she was now considering driving people to the polls and texting her friends to encourage them to attend other rallies in support of abortion rights.

“I’m out here trying to build a movement,” said Ms. Moser, who wore a red bandanna and held up a sign that read, “I will not go quietly back to the 1950s.”

In Brooklyn, thousands of abortion rights supporters gathered in Cadman Plaza Park before marching to Foley Square in Lower Manhattan. Volunteers offered snacks and signs with phrases like “Stand With Black Women.”

For some, protesting the draft opinion was not just about protecting the right to abortion.

Lillian Penafiel, 35, and her wife, Emi Penafiel, 44, worried about what a ruling would mean for marriage equality, L.G.B.T.Q. rights and voting rights.

“They’ve been very clear, especially what was written up, that our rights are going to be threatened as well, too, so that’s why we’re nervous,” Emi Penafiel said. “They’re coming after all of it.”

Madeleine Ngo reported from Washington, and Lola Fadulu from New York.