WASHINGTON — The Supreme Court appeared splintered on Wednesday during arguments in a major abortion case that could affect the lives of millions of American women.

The court’s four liberal justices were adamant that restrictions imposed by a Texas law on the state’s abortion providers served no medical purpose and could not pass constitutional muster. But two of the more conservative justices said there was little evidence that abortion clinics in Texas had closed or would close because of the law.

Justice Anthony M. Kennedy, who almost certainly holds the crucial vote, said it would help to know how many abortions could be performed in clinics if the law were allowed to become fully effective, and mused that it might be useful to return the case to the lower courts to develop more evidence.

The practical short-term consequences of Justice Kennedy’s suggestion would turn on what the Supreme Court does with a stay it issued in June, in which, by a 5-to-4 vote — with Justice Kennedy joining the court’s liberal wing — it temporarily blocked an appeals court’s ruling largely upholding the Texas law. Should the stay be lifted while the case proceeds, some 10 clinics might be forced to close in the meantime.