IN the abstract, most Americans apparently favor the right to end one's own life with the help of a doctor. But when they enter the quiet privacy of the voting booth, when they consider their own lives and especially those of their loved ones, the enormity of the question can bring a different result that touches on deep ethical concerns and religious beliefs.This is the lesson many analysts are drawing from Tuesday's election in Washington State, where voters rejected an initiative to allow physicians to participate in euthanasia. Those who supported the measure say the defeat came because of insufficient legislative safeguards to prevent abuse, which backers say will be remedied in similar efforts underway in California and Oregon and are being seriously discussed in a number of other states. "The public has thought it through and decided it wants it," says Jack Nicholl, campaign director for Californians Against Human Suffering, which last month started gathering signatures for the 1992 ballot. Mr. Nicholl and others note last week's national poll, sponsored by the Boston Globe and the Harvard School of Public Health, which showed that 64 percent of Americans favor physician-assisted suicide. Backers of such measures also point out that the Roman Catholic Church poured considerable money into blocking the Washington initiative, which happened to be listed alongside a controversial abortion-rights measure. But organized religion alone does not explain the opposition. Next to Oregon, Washington has the highest percentage of people who do not identify themselves with any organized religion. "Legalization of euthanasia poses a far deeper moral crisis than we may appreciate," observes Edmund Pellegrino, a physician and director of the Center for Advanced Study of Ethics at Georgetown University, in a paper written earlier this year. "It challenges us to define what it really means to be a physician. It forces us to define who we are, what we are, and what we want to be." In Washington, the issue was presented as one of "choice;" backers avoided the use of the word euthanasia. But here, too, deep moral and ethical questions are raised. Daniel Callahan, director of The Hastings Center (an independent research organization dealing with medical ethics), says that unlike suicide, euthanasia "should be understood as of its nature a social act." "We should not deceive ourselves into thinking of euthanasia ... as merely personal acts, just a slight extension of the already- established right to control our bodies and to have medical treatment terminated," he wrote in a recent issue of Commonweal magazine. "It is a radical move into an entirely different realm of morality: that of the killing of one person by another." There is also concern that although a mentally-competent adult has made the choice, others have control. In a forthcoming book on euthanasia in the Netherlands, Dr. Carlos Gomez of the University of Virginia Health Science Center warns that most cases go unreported and in hundreds of instances the patient in fact had not given permission. "Plenty of people are worried that a doctor or their relatives might dispatch them," says Arthur Caplan, director of the Center for Biomedical Ethics at the University of Minnesota. Still others say the issue cannot be separate from the fact that 30 million Americans do not have medical insurance. "People are beginning to see what this means, not only for the elderly but for those who are alone or impoverished and already short-changed by the health-care system," says Leon Kass, a medical doctor and professor of social thought at the University of Chicago. Like suicide, says Dr. Kass, euthanasia is often "an angry or anxious plea for help, born of fear of rejection or abandonment, or made in ignorance of available alternatives that could alleviate pain and suffering." Such requests, he says, must be seen as "incoherent." Dr. Kass, who is on leave at the American Enterprise Institute studying religious issues, also says the end of human life presents a chance to comprehend something profound about the individual and his or her family, a chance that would be lost if one ended one's own life. "The sad fact of mortality," he says, "is that it very often provides the opportunity to demonstrate how one lives and why one lives." "What I would hope is that one would see this crisis as an opportunity to rethink and reconsider exactly what we've been doing with the end of life - that there are all kinds of alternatives to mercy killing," Kass says. "Any suicide is a reminder of our failure to respond to an anguished plea for help and love, to embody as a community the commitment not to forsake one another," the Roman Catholic bishops of Oregon and Washington wrote in a recent pastoral letter. While Catholic and conservative Protestant denominations actively oppose suicide, a number of other faiths are now considering support for doctor-assisted euthanasia. Some, like the United Church of Christ and the Unitarian Universalist Association, have approved. "It's inevitable that we're going to keep talking about it," Mr. Caplan says. "It'll be a debate in our society for years."