The judges voted 5-4 early on Thursday to deny an emergency appeal from abortion providers and others that sought to block enforcement of the law.
Signed by Republican Governor Greg Abbott in May, the law prohibits abortions once medical professionals can detect cardiac activity, usually about six weeks and before most women know they are pregnant.
Abortion rights groups say such a ban has never been permitted in any state since the Supreme Court decided Roe v Wade, the landmark ruling that legalised abortion nationwide in 1973.
The Supreme Court said its ruling does not make any conclusions on the constitutionality of the Texas law and allows legal challenges to the legislation to move forward.
“In reaching this conclusion, we stress that we do not purport to resolve definitively any jurisdictional or substantive claim in the applicants’ lawsuit,” the majority said in the unsigned order.
“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, Justice Stephen Breyer, Justice Sonia Sotomayor and Justice Elena Kagan dissented.
Sotomayor called the majority’s decision “stunning”.
“Presented with an application to enjoin a flagrantly unconstitutional law engineered to prohibit women from exercising their constitutional rights and evade judicial scrutiny, a majority of justices have opted to bury their heads in the sand,” she said in her dissenting opinion.
“My administration is deeply committed to the constitutional right established in Roe v Wade nearly five decades ago and will protect and defend that right,” he said in a statement.
Julia Kaye, a lawyer with the American Civil Liberties Union’s Reproductive Freedom Project, told Al Jazeera there was “devastation and chaos” in Texas as a result of the ban.
“There are thousands of pregnant Texans who are sitting at their kitchen tables trying to crunch the numbers and figure out how they can possibly travel hundreds of miles out of state, in order to get time-sensitive medical care,” she said.
Communities of colour and low-income Texans would be hardest-hit by the new law, she added.
SOURCE: AL JAZEERA AND NEWS AGENCIES