Democratic presidential candidate, former Vice President Joe Biden speaks during the Human Rights Campaign dinner at Ohio State University, Saturday, June 1, 2019, in Columbus, Ohio.

Joe Biden, the front-runner in the Democratic presidential primary, said Thursday that he supports the repeal of the Hyde Amendment, which bars federal funding for most abortions.

"Women's rights and women's health are under assault like we haven't seen in the last 50 years," the former vice president said at an event in Atlanta. "If I believe healthcare is a right, as I do, I can no longer support an amendment that makes that right dependent on someone's zip code."

His comments come after he faced criticism this week for his support for the measure.

The Hyde Amendment bars federal funding for abortions — except in cases of rape or incest, or when the life of the mother is at risk.

Read more: Joe Biden says he still supports ban on federal funding of abortions, after apparent reversal

Thursday's announcement marks the second time in about a month that Biden has appeared to reverse his position on the Hyde Amendment.

At an event in South Carolina in early May, Biden told an ACLU volunteer that he supported abolishing the Hyde Amendment.

"It can't stay," Biden said at that time.

Then, on Wednesday, Biden's campaign said that the Delaware Democrat still supported the law, and had misheard the ACLU volunteer's question.

ACLU video on Twitter: Our volunteer Nina asked Joe Biden whether, as president, he would lift the Hyde amendment, which bans federal insurance coverage of abortion.

"He has not at this point changed his position on the Hyde Amendment," the campaign told CNBC Wednesday in a statement.

"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. 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," the statement said. The Roe v. Wade was the landmark Supreme Court ruling in 1973 that barred state abortion bans.

Biden's continued support for the Hyde Amendment sparked a unified uproar among his fellow Democrats and from reproductive rights groups, including Planned Parenthood, EMILY's List and NARAL.

In a statement Thursday, Biden's campaign said: "He has for many years as a United States Senator – as have many people – supported the Hyde Amendment."

But the campaign said that "he acknowledged tonight that the circumstances have changed."

"It's clear these folks will stop at nothing to get rid of Roe — and it's clear we have to be just as strong in our defense of it. He cannot justify leaving millions of people without access to the care they need, or the ability to exercise a constitutionally protected right," the statement said. "He believes health care is a right and he can no longer support an Amendment that makes it a right dependent on one's income or zip code."

Biden tweet: Women's rights and health care are under assault in a way that seeks to roll back every step of progress we've made over the last 50 years. If I believe health care is a right, as I do, I can no longer support an amendment that makes that right dependent on someone's zip code.

Biden's announcement in Atlanta is noteworthy. Georgia's Republican governor last month signed into law a ban on abortions after a doctor can detect a fetal heartbeat, or about six weeks.

Companies including Disney, Netflix, and CNBC parent NBCUniversal have said that if the law goes into effect, it could influence their decisions regarding working in the state.