The judges voted 5-4 to deny an emergency appeal from abortion providers and others that sought to block enforcement of the law, which came into effect on Wednesday.
Despite the ruling from the US' highest court, this is not likely to be the last word on whether the law can stand because other challenges to it can still be brought.
The Texas law, signed by Republican Governor Greg Abbott in May, prohibits abortions once medical professionals can detect cardiac activity in the foetus, usually around six weeks and before many women know they are pregnant.
It is the strictest law against abortion rights in the United States since the high court’s landmark Roe v Wade decision in 1973.
It is also part of a broader push by Republicans nationwide to impose new restrictions on abortion. At least 12 other states have enacted bans early in pregnancy, but all have been blocked from going into effect.
The Supreme Court's order declining to halt the Texas law came just before midnight on Wednesday. The majority said those bringing the case had not met the high burden required to stop the law from coming into force.
“In reaching this conclusion, we stress that we do not purport to resolve definitively any jurisdictional or substantive claim in the applicants’ lawsuit," the nine judges said in a statement.
"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,” the unsigned order said.
Reacting to the vote, President Joe Biden said the "extreme" law violates women's constitutional rights.
"The Texas law will significantly impair women’s access to the healthcare they need, particularly for communities of colour and individuals with low incomes," he said.
"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." Chief Justice John Roberts dissented along with the court’s three liberal justices. Each of the four dissenting justices wrote separate statements expressing their disagreement with the majority.
Chief Justice Roberts noted that while the majority denied the request for emergency relief “the Court’s order is emphatic in making clear that it cannot be understood as sustaining the constitutionality of the law at issue".
The vote in the case underscores the impact of the death of the liberal Justice Ruth Bader Ginsburg last year and then-president Donald Trump’s replacement of her with conservative Justice Amy Coney Barrett. Had Ms Ginsburg remained on the court there would likely have been five votes to halt the Texas law.
Justice Sonia Sotomayor called her conservative colleagues’ 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 wrote.
The law does not make exceptions for rape or incest. However, it does allow abortions if the pregnancy could endanger the mother’s life or cause “substantial and irreversible impairment of a major bodily function”.
Texas lawmakers wrote the law to evade federal court review by allowing private citizens to bring civil lawsuits in state court against anyone involved in an abortion, other than the patient. Other abortion laws are enforced by state and local officials, with criminal sanctions possible.
In contrast, Texas’ law allows private citizens to sue abortion providers and anyone involved in facilitating abortions. Among other situations, that would include anyone who drives a woman to a clinic to get an abortion. Under the law, anyone who successfully sues another person would be entitled to at least $10,000 (approximately £7,260).
President Biden called this detail "outrageous", saying the it "deputises private citizens to bring lawsuits against anyone who they believe has helped another person get an abortion, which might even include family members, health care workers, front desk staff at a health care clinic, or strangers with no connection to the individual".
In her dissent, Justice Elena Kagan echoed the president's comments. She described the law as “patently unconstitutional,” highlighting that it allows “private parties to carry out unconstitutional restrictions on the state’s behalf.”
Justice Stephen Breyer said a “woman has a federal constitutional right to obtain an abortion during” the first stage of pregnancy.
After a federal appeals court refused to allow a prompt review of the law before it took effect, the measure’s opponents sought Supreme Court review.
In a statement early on Thursday after the high court’s action, Nancy Northup, the head of the Centre for Reproductive Rights, which represents abortion providers challenging the law, vowed to “keep fighting this ban until abortion access is restored in Texas.”
“Texas politicians have succeeded for the moment in making a mockery of the rule of law, upending abortion care in Texas, and forcing patients to leave the state - if they have the means - to get constitutionally protected healthcare.
"This should send chills down the spine of everyone in this country who cares about the constitution,” she said.
Texas has long had some of the nation’s toughest abortion restrictions, including a sweeping law passed in 2013. The Supreme Court eventually struck down that law, but not before more than half of the state’s 40-plus clinics closed.
Even before the Texas case arrived at the high court the justices had planned to tackle the issue of abortion rights in a major case after the court begins hearing arguments again in the autumn. That case involves the state of Mississippi, which is asking to be allowed to enforce an abortion ban after 15 weeks of pregnancy.