WASHINGTON, Jan. 8 -- By the end of two intense hours of Supreme Court debate today over whether the Constitution gives terminally ill people a right to doctor-assisted suicide, it was clear that the Justices were fascinated by the issue and deeply engaged by the arguments -- but at the same time eager to keep the Court out of yet another momentous question of life and death.

''Everything you've said, it seems to me, could go on in a legislative chamber,'' Justice Ruth Bader Ginsburg told Kathryn L. Tucker, a Seattle lawyer arguing that Washington State's law against assisted suicide violates the due process rights of terminally ill patients who want a doctor's help in dying in a ''humane and dignified manner.''

Justice David H. Souter asked Ms. Tucker, ''Why shouldn't we conclude that as an institution, we are not in a position to make the judgment you want us to make?'' Referring to the consequences of opening the door to doctor-assisted suicide, Justice Souter said, ''It would just be guesswork.''

Although the doctrinal jargon of due process and equal protection was sprinkled throughout the argument, this was a Supreme Court session notable for the proportion of plain English that was spoken. The Justices wanted theory, but they were also hungry for facts. They sat as judges but appeared to feel themselves very much participants, on a human level, in a far-reaching societal debate.

''This is an issue every one of us faces, young and old, male and female, whatever it might be,'' Justice Sandra Day O'Connor said at one point. And Justice Ginsburg, whose mother died of cervical cancer at the age of 47, said, ''Most of us have parents and other loved ones who have been through the dying process, and we've thought about these things.''

Yet this was not a policy forum, but a legal argument before a Supreme Court that is years past any desire to find itself at the frontiers of constitutional law. The lawyers asked the Justices to weigh terminally ill patients' interest in avoiding uncontrollable pain against the states' interest in protecting life. But the Justices' concern often appeared to be the Court's own interest in avoiding the vulnerability that comes from folding into the Constitution a claimed right that society has not fully embraced.

Justice Souter, observing at one point that ''20 years ago, we weren't even reading about this,'' said, ''Maybe the Court should wait until it could know more.'' That led Justice Ginsburg to ask whether it was merely a question of ''just waiting'' -- the issue, she said, might be whether the Court should endorse the concept of doctor-assisted suicide ''now or ever.''

''This case raises the basic question of who decides,'' Justice Ginsburg said. ''Is this ever a proper question for courts to decide?''

The Justices are scheduled to rule on the matter before the end of the term in late June or early July. The Court agreed three months ago to take up the issue after two Federal appeals courts, ruling almost simultaneously last spring in suits brought by doctors and their terminally ill patients, struck down criminal prohibitions against assisted suicide in the states of Washington and New York.

These were the first Federal appellate rulings on the question, and both states filed appeals.

Dennis C. Vacco, New York's Attorney General, argued the New York appeal, Vacco v. Quill, No. 95-1858. William L. Williams, senior assistant attorney general of Washington, argued that state's appeal, Washington v. Glucksberg, No. 96-110.

The Clinton Administration entered the cases on behalf of the states. Solicitor General Walter Dellinger argued in both cases today.

In contrast to the states' lawyers, who urged the Court to find that there was no constitutional right at stake, Mr. Dellinger urged the Justices to recognize that terminally ill people have a ''liberty interest'' in not having the state prevent their relief, through doctor-assisted suicide, from ''severe pain and suffering.'' However, he said, the states' interest in ''affirming the value of life'' and protecting vulnerable patients should be given even greater weight, and the prohibitions should be upheld.

''The systemic dangers are dramatic'' in a society that allows doctor-assisted suicide, Mr. Dellinger said. ''The least costly treatment for any illness is lethal medication,'' he added.

The distinction between the Clinton Administration's view and the states' position is more than merely academic. A finding by the Court that a ''liberty interest'' is at stake would sweep the question of regulating doctor-assisted suicide within the ambit of the 14th Amendment, which prohibits the states from depriving ''any person of life, liberty, or property, without due process of law.'' It would commit the Court to a kind of ongoing constitutional supervision of the question.

How to define the substantive reach of the due process clause is perhaps the most profound constitutional debate on the Court today, and the Justices are likely to be sharply divided over this approach even if all or most agree that the state laws should be upheld.

''A liberty interest tosses this whole matter back to this Court,'' said Justice Antonin Scalia, the Court's most outspoken opponent of announcing new due process rights. His tone was sarcastic.

''I suppose proclaiming a liberty interest is cost-free,'' he said. Several months ago, in a speech at Catholic University here, Justice Scalia said it was ''absolutely plain that there is no right to die.''

He challenged Laurence H. Tribe, a Harvard Law School professor who was arguing on behalf of the plaintiffs in the New York case that ''when facing imminent and inevitable death,'' a person has the right ''not to be forced to be a creature of the state'' and to ''have some voice'' in the way the ''final chapter of life'' unfolds.

''This is lovely philosophy,'' Justice Scalia said. ''Where is it in the Constitution?''

In a 1990 decision, Cruzan v. Missouri Department of Health, the Court used a liberty-interest analysis to conclude that people have a right to avoid unwanted medical treatment, including treatment that is keeping them alive.

The meaning and relevance of that precedent were much discussed today.

In striking down New York's prohibition against assisted suicide, the United States Court of Appeals for the Second Circuit ruled that it violated the 14th Amendment's guarantee of equal protection because it allowed the state to treat differently two categories of terminally ill people: those on life-support systems, who have a right under state law as well as under the Cruzan decision to have the equipment withdrawn, and those who have no machinery to disconnect and who are denied the ability to hasten their deaths by prescribed medication.

Mr. Tribe said the distinction drawn under New York law not only was arbitrary, but also was based on a ''fantasy'' that there was a sharp demarcation in medical terms between the two groups of patients. Justice Anthony M. Kennedy said the distinction was between active intervention in the case of assisted suicide and ''nature taking its course'' for those who die after the withdrawal of equipment, but to that argument Mr. Tribe said, ''None of these patients is in a state of nature.''

Mr. Tribe said that terminally ill patients whose life-support equipment is removed are often placed in a deeply sedated condition, in which they die under medical supervision.

Attorney General Vacco, arguing the state's appeal, said the Constitution's equal protection guarantee was ''not implicated at all'' in the way New York treats the two categories of terminally ill patients because those who withdraw life support are ''merely asserting the right to be let alone,'' while those who seek assisted suicide want a third party's help in killing themselves.

In the Washington case, the United States Court of Appeals for the Ninth Circuit, which sits in San Francisco, based its decision not on equal protection, but on the due process guarantee, finding a ''liberty interest in choosing the time and manner of one's death.''

Mr. Williams, representing Washington, said the decision denied the state the ability to maintain ''a clear line between physicians as healers and curers and physicians as instruments of the death of their patients.''

Ms. Tucker, representing the plaintiffs in the Washington case, said judicial intervention was needed because ''ours is a culture of denial of death'' in which the ordinary political process could not work to protect the rights of terminally ill people. Referendums to establish doctor-assisted suicide narrowly failed in Washington in 1991 and in California in 1992; in a 1994 referendum, voters in Oregon approved an assisted-suicide law, which has been blocked in Federal court.

Chief Justice William H. Rehnquist asked Ms. Tucker why her clients' constitutional claim included a doctor's assistance in suicide rather than simply the right to kill themselves.

Ms. Tucker replied that her terminally ill clients -- all of whom died early in the litigation -- wanted ''a peaceful, humane, dignified death'' through medication for which a doctor is the ''gatekeeper.''

Justice Scalia asked why Ms. Tucker had limited her constitutional claim to people with terminal illnesses. Why, he asked, would someone faced with 10 years of ''terrible suffering'' not have a similar right?

The dying patient ''does not have a choice between living and dying,'' Ms. Tucker replied. ''The dying process has begun,'' and the patient has a choice only of controlling the manner of death.

''I have to tell you, the dying process of all of us has begun,'' Justice Scalia said.