By Eric Bradner, CNN

(CNN) -- Joe Biden still supports a prohibition on federal funds being used for abortion, except in cases of rape, incest or when the life of the mother is at stake, his campaign said Wednesday.

The former vice president's continued support for the Hyde Amendment, first reported by NBC News, positions him to the right of his major 2020 Democratic rivals who have called for its repeal -- and quickly drew rebukes from those candidates. It was the first time Biden's Democratic rivals had all at once taken aim at the party's front-runner for the 2020 nomination.

Massachusetts Sen. Elizabeth Warren said while campaigning in Indiana that the Hyde Amendment denies abortion rights to "people who don't have money," and said that's "just not right."

"No woman's access to reproductive health care should be based on how much money she has. We must repeal the Hyde Amendment," California Sen. Kamala Harris tweeted.

New York Sen. Kirsten Gillibrand tweeted: "Reproductive rights are human rights, period. They should be nonnegotiable for all Democrats."

Vermont Sen. Bernie Sanders pledged on Twitter to repeal the Hyde Amendment. Former Rep. Beto O'Rourke tweeted, "No matter your income or where you live, every woman should have access to health care including abortion."

Democrats broadly oppose the Hyde Amendment because they say it creates an economic disparity in access to abortions: Lower-income women who rely on government-funded programs like Medicaid don't have the same access as wealthier women who can afford them. It also interjects the federal government into what most Democrats say should be decisions made by a woman and her doctor.

Hillary Clinton opposed the Hyde Amendment during the 2016 presidential campaign, and Democrats adopted that opposition as part of the party's official platform.

Other Democrats running for president this year have backed legislation that would overturn the Hyde Amendment. Sens. Gillibrand, Harris, Amy Klobuchar, Warren, Sanders and Cory Booker are among the co-sponsors of a bill Illinois Sen. Tammy Duckworth introduced in March.

The debate over abortion rights has heated up as largely Republican states enact a series of restrictions that could ultimately lead to the Supreme Court revisiting Roe v. Wade. That debate is now at the forefront of the 2020 race, with Democratic candidates pledging to push measures to protect a woman's right to an abortion.

The Biden campaign's affirmation of his support for the Hyde Amendment comes despite Biden personally committing twice to oppose it when asked by ACLU volunteer after an early May campaign stop in South Carolina.

"Yes," Biden said during the exchange, video of which the ACLU posted on Twitter. When the woman asks again, Biden responds that "it can't stay."

Biden's campaign said in an emailed statement Wednesday that he "misheard the woman on the ropeline and thought she was referring to the Mexico City rule" -- a policy that prohibits federal funding for non-governmental organizations that offer or refer patients for abortions or advocate for expanded access to abortion rights. That policy is broadly opposed by Democrats and was repealed by Presidents Bill Clinton and Barack Obama, only to be re-instated by George W. Bush and Donald Trump.

"He has not at this point changed his position on the Hyde Amendment. The Hyde Amendment does not prevent organizations in the US that provide lifesaving health care services for women from receiving the federal funding they need," Biden's campaign said. "But given the current draconian attempts to limit access to abortion, if avenues for women to access their protected rights under Roe V Wade are closed, he would be open to repeal."

Biden also faced sharp criticism from women's health care and pro-abortion rights organizations.

Ilyse Hogue, the president of NARAL Pro-Choice America, said in a statement there is "no political or ideological excuse for Joe Biden's support for the Hyde Amendment, which translates into discrimination against poor women and women of color plain and simple."

"Differentiating himself from the field this way will not earn Joe Biden any political points and will bring harm to women who are already most vulnerable," Hogue said.

Kelley Robinson, the executive director of Planned Parenthood Action Fund, said in a statement on Wednesday: "Supporting Hyde isn't good policy or politics. We strongly encourage Joe Biden to speak to the people whose lives are impacted by this discriminatory policy and reevaluate his position."

Biden is a devout Catholic who early in his Senate career voted in favor of measures that would restrict abortion rights. He later became a defender of Roe v. Wade, the Supreme Court decision that legalized abortion nationwide.

"I refuse to impose my religious beliefs on other people. Women should control their own bodies -- period," he wrote in a May 21 email to supporters criticizing largely Republican-controlled states that had recently enacted strict new limits on abortion rights.

Biden spokesman Jamal Brown said that the former vice president "has fought vigorously to protect a woman's right to choose and against measures criminalizing abortion."

"Vice President Biden believes we must protect the progress we've made and has called on codifying the decision in Roe to ensure this choice remains between a woman and her doctor," Brown said.