WASHINGTON (CNN) -- The U.S. Supreme Court takes on two high-profile abortion cases this week, refocusing attention on one of the court's biggest judicial and social conflicts. The confirmation battle over court nominee Samuel Alito is likely to center to a large extent on these cases and his own views on abortion rights."For better or worse, the issue of abortion seems to be the one on which the Supreme Court is judged more than any other," said Edward Lazarus, a former clerk to Justice Harry Blackmun, the author of the 1973 Roe v. Wade ruling legalizing abortion. (Watch the cases being heard) Parental notification an issueThe most-watched case, to be argued Wednesday, deals with a 2003 New Hampshire law that would make it illegal for an abortion to be performed on a minor unless a parent or legal guardian had been notified in writing 48 hours in advance. The only exception would be if the procedure was necessary to prevent the minor's death.A federal appeals court found that exception is not broad enough, ruled it unconstitutional and blocked it from taking effect."This law does seem to be crafted to demand some kind of Supreme Court review," Lazarus said. "It really goes to this question of how much you have to protect the life versus the health of the mother. That comes from the initial ruling in Roe v. Wade. It's not an issue that has been settled at the court."Among those who would have been affected by the law is a New Hampshire woman who recently spoke with CNN. She had an unplanned pregnancy as a teenager. "I decided it was best for me to have an abortion because I did not want to be a parent at that point in my life," she said.Now 20 and a college senior, she is speaking out against the state statute."These laws are only about eroding access to abortion," she told CNN. "If you want to talk to your parents, you can do that, but if the state steps in and tells you that you have to do that to protect your reproduction, it is very disappointing."Poll: Americans favor limits, oppose banAt least 33 states have parental notification laws. A new CNN/USA Today/Gallup Poll found 69 percent of Americans surveyed favored a law requiring minors to get parental consent for an abortion; 28 percent opposed such a requirement. But 61 percent did not support a constitutional amendment to ban abortion, consistent with polls over the years on the question. Thirty-seven percent favored such an amendment."In a narrow sense, the public is very concerned about parental notification and other restrictions on the right to abortion," said Thomas Goldstein, an appellate attorney who has argued many cases before the justices."More broadly, this case is really a bellwether for where the Supreme Court is going to go in terms of limiting Roe v. Wade, and potentially eventually overruling it."New Hampshire argues separate provisions -- which it calls "safety valves" -- provide for a health exception by allowing doctors to seek an emergency judicial waiver of the notification requirement for non-life-threatening health issues.Legal scholars say the case will turn on whether New Hampshire's law represents an "undue burden" on women seeking abortions. The high court has followed that standard when deciding whether such laws are too restrictive. Supporters of the state say the law in question falls far below that."Should parents be notified if a minor's going to have an abortion? Of course our answer is that they should be notified," said Jay Sekulow, chief counsel of the American Center for Law and Justice. "You're not talking about parental consent, you're just talking about notification," he added. "In high school, a kid can't even have an aspirin without getting a parental slip, so the idea that they could have an abortion procedure without telling the parents that it's about to happen just seems to be outrageous."Decision has national impactA ruling in the case will have national implications for a variety of laws dealing with access to abortion.The last time the high court intervened in an important abortion-related case was in 2000, when it threw out a Nebraska law banning a controversial late-term procedure opponents called "partial-birth" abortion. Since Roe v. Wade, various states have tried to place restrictions and exceptions on access to the procedure, prompting a string of high court "clarifications" on the issue over the years.In the New Hampshire case, the justices will delve into a subtle but potentially monumental argument over what legal standard should be applied when courts review abortion laws. A federal judge in the case last year allowed courts to block the law while the case was appealed.That legal standard is important because the "partial-birth" issue is now back before the high court. Both the federal government and several states have prohibited the late-term abortion procedure, but the ban has not yet been allowed to go into effect. President Bush signed the 2003 Partial-Birth Abortion Ban Act, but it contains no health exception.Appeals could reach the high court within weeks, and if the justices accept the cases, they could be heard next spring, with two new members on the bench.Protest restrictions revisitedA separate case argued an hour before the New Hampshire appeal will tackle another contentious product of the abortion debate: protests outside medical and reproductive clinics.Courts have been wrestling with this issue for two decades, prompting one lawsuit and appeal after another. Complicating matters have been conflicting rulings over the years from the Supreme Court.As far back as the mid-1980s, anti-abortion groups began offering "classes" on abortion-clinic protest strategies. The high court in the past has ruled on protest guidelines: specifying distance from the clinics and what kind of conduct is permissible.The justices will revisit their ruling of two years ago that said protesters cannot be prosecuted simply for harassing patients and staff, blocking doors and other disruptive behavior. The court will clarify whether federal laws against racketeering and extortion can be used against those who, according to the official court filing, organize "sit-ins and demonstrations that obstruct public's access" to medical clinics.Abortion rights supporters say those laws were the only solution to what they call dangerous, often violent behavior aimed against those seeking or providing the medical procedure."There are protections now through some laws to make sure that people are safe, coming and going, from reproductive health providers," said Karen Pearl, interim president of Planned Parenthood, which operates 850 affiliated health centers. "This case is important because it puts again, into our law, one more way to stop that kind of violence and domestic terrorism against providers and against patients who need services."This group and others filed suit in federal court more than a decade ago.Operation Rescue, one of the plaintiffs in the case, says this is about free speech and the right of assembly.