Nearly one in five voters in battleground states says that President Biden is responsible for ending the constitutional right to abortion, a new poll found, despite the fact that he supports abortion rights and that his opponent Donald J. Trump appointed three Supreme Court justices who made it possible to overturn Roe v. Wade.

Trump supporters and voters with less education were most likely to attribute responsibility for abortion bans to Mr. Biden, but the misperception existed across demographic groups. Twelve percent of Democrats hold Mr. Biden responsible, according to New York Times/Siena College polls in Arizona, Georgia, Michigan, Nevada and Wisconsin and a Times/Philadelphia Inquirer/Siena poll in Pennsylvania.

“I think the buck stops with him, so he had the ability to fight that, and that’s not what I’m hearing that he did,” said Terri Yonemura, 62, an abortion rights supporter in Las Vegas who said she would not vote for Mr. Trump, but is unsure about Mr. Biden, so may not vote at all.

Abortion has been a mobilizing issue for Democrats in recent elections, and the confusion among a segment of voters presents both a challenge and an opportunity for Mr. Biden, who trailed Mr. Trump by six points in the survey overall.

“This group is a pickup opportunity for Democrats,” said Celinda Lake, a Democratic pollster who regularly surveys voters about abortion. The bigger challenge, she said, is that many voters do not understand Mr. Trump’s stance on the issue. “He has intentionally kept it vague. But when we show voters his statements in his own words, that is enough to persuade them.”

The message has become central in Mr. Biden’s campaign. It has been running ads linking Mr. Trump to abortion bans. One says explicitly: “Trump did this.” Vice President Kamala Harris has been highlighting the issue in public appearances and interviews.

Many voters who held Mr. Biden responsible said they simply didn’t pay close attention to politics or government affairs. For some, the confusion came from the fact that the Supreme Court’s Dobbs decision happened while Mr. Biden was president.

DeLana Marsh, 30, of Holly Springs, Ga., supports abortion rights and opposes a new Georgia law that bans abortions after six weeks of pregnancy: “I don’t think a group of men should be able to decide that for us.”

But she said she was under the impression that Mr. Biden was responsible because it happened during his presidency, and she believed his age prevented him from closely tracking such events.

Other voters said Mr. Biden hadn’t done enough to stop state abortion bans. (He has criticized the Dobbs decision and enacted certain federal policies to support abortion rights, and does not have the authority to reverse state laws.)

“There should be no restrictions on abortion whatsoever,” said Ana Juarez Ramirez, 18, of Nogales, Ariz. Yet he says Mr. Biden made empty promises on many issues, including abortion.

“Biden did not fully criticize or condemn the taking away of people’s rights,” he said. Still, he plans to vote for Mr. Biden, mostly because, he said, “I don’t even want to think about voting for Trump.”

Overall, voters in battleground states say they trust Mr. Biden more than Mr. Trump to handle the issue of abortion, largely unchanged since last November. But even so, about 6 percent of Democrats, including many who want abortion to remain legal, say they trust Mr. Trump more to handle the issue.

Some voters said they did not believe that Mr. Trump actually opposed abortion rights. Among Republicans, only about four in 10 held him responsible for Dobbs, a figure that may reflect partisanship as well as confusion. But on some aspects of the abortion debate, Mr. Trump has sent mixed signals..

Mr. Trump has clearly claimed responsibility for the decision: “I was able to kill Roe v. Wade,” he posted last year, and last month reiterated that he was “proudly the person responsible” for doing so.

Yet recently, he wouldn’t commit to a position on a national abortion ban and said he would allow states to prosecute women who violated abortion restrictions. But several weeks earlier, he said he believed abortion law should be left to the states and include exceptions.

Christine Valenti, 72, is a Republican from Wisconsin and two-time Trump voter who says abortion should be mostly legal and that women in states with bans should be able to travel to another state to get one.

But she said that Mr. Trump’s recent statements on leaving abortion up to the states assured her that his views were in line with hers. And she said Mr. Biden hadn’t done enough to support abortion rights: “He doesn’t say much about it anymore. He’s our president, but he doesn’t say a lot, period, about anything.”

Ultimately, though, she said the economy was her more pressing concern. When voters were asked for the one issue most important to them in the election, the largest share of respondents, 21 percent, said the economy. Abortion and immigration were next, with just over 10 percent saying each was most important.

The New York Times/Philadelphia Inquirer/Siena College poll of Pennsylvania was funded by a grant from the Lenfest Institute for Journalism. The poll was designed and conducted independently from the institute.