Here's Linda Greenhouse's analysis of the argument:
Justices across the ideological spectrum appeared inclined to send the case back to the federal appeals court that had declared the law unenforceable in all respects, and to instruct that court to render a narrower ruling. Such a ruling would permit the law to take effect except when a doctor had certified that an immediate abortion - without either notifying a parent or seeking approval from a judge, an option known as a judicial bypass - was necessary to preserve a girl's health....I consider Stevens's question a devastating one, showing the bad faith of the state legislature, a deliberate hostility to the constitutional right the Supreme Court has recognized.
Attorney General Kelly A. Ayotte of New Hampshire, who brought the appeal of the lower court's ruling, asserted in her argument that under New Hampshire's general health law, a doctor performing an emergency abortion would have a legal defense in any event, based on the state's general law regarding medical practice. Ms. Ayotte said she was prepared to issue a formal opinion to that effect if the occasion arose.
The attorney general's position left Justice Ginsburg unsatisfied. "That's the real problem here for the doctor who's on the line," she said. "I think a lawyer who cares about his client would say 'defense' is not what we want, what we want is that there is no claim; not that you have to put up a defense and maybe the attorney general will give us a letter saying that we come under that defense."
Justice John Paul Stevens reminded Ms. Ayotte that the sponsors of the parental notice law in the New Hampshire Legislature had rejected including a medical exception. "When you have legislative history that suggests that the Legislature considered this very defense and rejected it in the statute, would then that give some concern?"
Ms. Ayotte replied that while "there certainly was some indication that the Legislature did not want a general health exception," sponsors did not intend to leave pregnant teenagers unprotected in emergencies.
UPDATE: Listen to the whole oral argument here or download here. I'm surprised at how tempestuous the argument gets with the Solicitor General, who aggressively talks over a Justice more than once.