WASHINGTON — President Biden on Thursday excoriated the Supreme Court’s refusal to block a Texas law that prohibits abortion after six weeks, saying that it “unleashes unconstitutional chaos” against women and vowing that his administration would investigate how the federal government could protect existing constitutional abortion rights.

In a statement, the president said that he had directed a gender-focused policy council in the White House, the Office of the White House Counsel and the Health and Human Services and the Justice Departments to “launch of a whole-of-government effort” to respond to the court’s decision.

“Complete strangers will now be empowered to inject themselves in the most private and personal health decisions faced by women,” Mr. Biden said. “This law is so extreme it does not even allow for exceptions in the case of rape or incest.”

The Texas law, known as S.B. 8, makes it difficult for officials to enforce Roe v. Wade, a landmark decision that establishes the right to abortion until the point of fetal viability, the point at which fetuses can sustain life outside the womb, or about 22 to 24 weeks into a pregnancy.

The law allows anyone to bring a court case against anyone else involved in obtaining an abortion — including taxi drivers, receptionists or relatives who provide money for the procedure — making it difficult for abortion rights activists to challenge its restrictions in court. The law also further incentivizes plaintiffs to police women who are seeking the procedure by offering $10,000, in addition to legal fees, if they win in court.

The law, as Mr. Biden and Vice President Kamala Harris pointed out in statements this week, will disproportionately affect women of color: About 70 percent of abortions in Texas in 2019 were provided to women of color, according to the Guttmacher Institute.

“Patients in Texas will now be forced to travel out of state or carry their pregnancy to term against their will,” Ms. Harris said on Wednesday.

The Supreme Court’s vote to let the law stand was delivered with a 5-to-4 majority late Wednesday evening. In the unsigned majority ruling, justices sought to make clear that the Texas law was still subject to stronger cases against it. But even that, Mr. Biden said, would “allow millions of women in Texas in need of critical reproductive care to suffer while courts sift through procedural complexities.”

Siding with the court’s liberal wing, Chief Justice John G. Roberts Jr. wrote that he would have blocked the law as appeals moved forward.

“The statutory scheme before the court is not only unusual, but unprecedented,” he wrote. “The legislature has imposed a prohibition on abortions after roughly six weeks, and then essentially delegated enforcement of that prohibition to the populace at large. The desired consequence appears to be to insulate the state from responsibility for implementing and enforcing the regulatory regime.”

So far, the White House has offered few details on what the president or those in his administration could actually do to defang S.B. 8 or protect Roe v. Wade. In its next term, which starts in October, the Supreme Court will take up a separate case to decide whether it should be overruled.

Jen Psaki, the White House press secretary, said Wednesday that the president would “continue to call for the codification of Roe,” adding that the Texas law “highlights even further the need to move forward on that effort.”

Other Democrats vowed to take up measures to protect abortion rights. Speaker Nancy Pelosi said in a statement that S.B. 8 “delivers catastrophe to women in Texas,” and said that the House of Representatives would take up the Women’s Health Protection Act, which would provide new protections for abortion access, when lawmakers return back from their summer recess. It is unclear if that measure would garner the 60 votes needed to overcome a potential filibuster and be considered in the Senate.