The decision in Texas could end up shaking the foundation of the American political system as many Democrats want the White house to go to all lengths to overturn the decision, ITV News Presenter Tom Bradby reports
He also directed federal agencies to do what they can to “insulate women and providers” from the impact.
The Texas law, which came into effect on Wednesday, bans abortions once medical professionals can detect cardiac activity, usually around six weeks of pregnancy, before most women know they are pregnant.
It was signed by Republican Governor Greg Abbott in May.
Abortion providers and others that sought to block the law put in an emergency appeal but the Supreme Court voted 5-4 to deny it.
Mr Biden said his administration will launch a “whole-of-government effort to respond to this decision” and look at “what steps the federal government can take to ensure that women in Texas have access to safe and legal abortions as protected by Roe.”
He said women should be protected from “the impact of Texas’ bizarre scheme of outsourced enforcement to private parties.”
Mr Biden, who has come under pressure from Democrats to expand the size of the Supreme Court, has ordered a review of the court that is due next month.
The Texas law is the strictest law against abortion rights in the United States since the high court’s landmark Roe v Wade decision in 1973.
At least 12 other states have tried to ban abortions early in pregnancy, but all attempts have been blocked.
The Supreme Court's order, which came just before midnight on Wednesday, read: “In reaching this conclusion, we stress that we do not purport to resolve definitively any jurisdictional or substantive claim in the applicants’ lawsuit.
"In particular, this order is not based on any conclusion about the constitutionality of Texas’s law, and in no way limits other procedurally proper challenges to the Texas law, including in Texas state courts.”
Chief Justice John Roberts and the court’s three liberal justices - Justice Stephen Breyer, Justice Sonia Sotomayor and Justice Elena Kagan - voted against the legislation.
Texas politicians wrote the law to evade federal court review by allowing citizens to bring civil lawsuits against anyone involved in an abortion, other than the patient.
Other abortion laws are enforced by state and local officials, with criminal sanctions possible.
After a federal appeals court refused to review the law, opponents brought an appeal to the Supreme Court.
Texas has had some of the nation’s toughest abortion rules, including a sweeping abortion ban in 2013. The Supreme Court eventually struck down that law, but only after more than half of the state’s 40-plus clinics closed.
The Supreme Court will be hearing a similar case in autumn where the state of Mississippi is planning to enforce an abortion ban after 15 weeks of pregnancy.