One of the court's six conservatives, Chief Justice John Roberts, joined its three liberals in dissent.
"The court’s order is stunning," liberal Justice Sonia Sotomayor wrote in a dissenting opinion.
"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."
In an unsigned explanation, the court's majority said the decision was "not based on any conclusion about the constitutionality of Texas’s law" and allowed legal challenges to proceed.
The decision illustrates the impact of former Republican President Donald Trump's three conservative appointees, who have tilted the court further right. All were in the majority.
Source: Austin American-Statesman
The law would amount to a near-total ban on the procedure in Texas, as 85 per cent to 90 per cent of abortions are obtained after six weeks of pregnancy, and would probably force many clinics to close, abortion rights groups said.
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.
Texas is among a dozen mostly Republican-led states to ban the procedure once a foetal heartbeat can be detected, often at six weeks and sometimes before a woman realises she is pregnant.
Courts have blocked such bans, citing Roe v. Wade.
The court's action over the Texas ban could foreshadow its approach in another case over a 15-week ban by Mississippi in which the state has asked the justices to overturn Roe v. Wade.
The court will hear arguments in the term beginning in October, with a ruling due by the end of June next year.