Kennedy, who often casts the deciding vote in close cases, at one point suggested sending the case back to a lower court for further evidence to be introduced on the law's impact, which could mean the case might not be resolved for years.
With the court one member short after the Feb. 13 death of Justice Antonin Scalia, the court could be split 4-4.
A 4-4 decision would let stand a lower-court ruling that affirmed the Texas law but no nationwide legal precedent would be set on whether other states can enact similar measures.
If Kennedy, a conservative, sides with the four liberals, the court could strike the law down.
The court's four liberals, led by Justice Ruth Bader Ginsburg, indicated hostility to the law.
Justices on the court's conservative wing were more sympathetic to Texas, raising questions about whether the law had closed down as many clinics as abortion providers had said.
The state contends the law, passed in 2013 by the Republican-controlled Texas legislature and signed by a Republican governor, protects women's health.
The abortion providers who have challenged it assert that it is aimed at shutting down their clinics.
Scalia's death means the court no longer has five conservatives who might support more restrictive abortion regulations nationwide.
Hundreds of activists on both sides of the issue gathered outside the courthouse on a chilly, blustery day.
The Supreme Court legalized abortion more than four decades ago with its landmark ruling in the 1973 Roe v. Wade case.
But abortion remains a disputed issue in the United States, as it does in many countries, and some states have passed laws aiming to place a variety of restrictions on a woman's ability to terminate a pregnancy.
The Texas law requires abortion doctors to have "admitting privileges," a type of formal affiliation, at a hospital within 30 miles (48 km) of the clinic.
Abortion providers say the provision already has forced clinics to close because such an affiliation is hard for clinic doctors to obtain. Read more: