Justice Stevens, in his dissenting opinion, uses a form of reductio ad adsurdum argument to show that the primary premise of the majority opinion is untenable.
A reductio argument works by showing that a given premise leads to an absurd conclusion - and thus the initial premise is unacceptable.
Justice Stevens' version works by showing that the primary premise of the majority opinion leads to two (or more) conclusions - which contradict one another. If a starting premise leads to contradiction - then there's something wrong with the premise.
To paraphrase:

Premise: In the majority view, chronically incompetent persons are not persons in the meaning of the Constitution - ie. they do not have constitutionally cognizable interests.

Conclusion: Therefore, the State of Missouri must protect the (physiological) life of Nancy Cruzan - even against the wishes of her parents, "quality of life" arguments to the contrary, etc.

Justice Stevens, however, points out that the premise can also lead to a second conclusion - one that directly contradicts the conclusion drawn in the majority opinion:

Premise: Chronically incompetent persons are not persons in the meaning of the Constitution - ie. they do not have constitutionally cognizable interests.

Conclusion: Therefore, State X can refuse life-support treatment - indeed, order lethal injections - for chronically incompetent persons (because they are not persons in the constitutional sense, and enjoy no rights, e.g., the right to life).

Again, if the starting premise - "chronically incompetent persons are not persons in the meaning of the Constitution - ie. they do not have constitutionally cognizable interests" - can be shown to lead to two directly contradictory conclusions, the premise is untenable.
(In addition to his logical critique, Justice Stevens goes on to point out that a law based on such a premise is useless: because it leads to such contradictory conclusions, it is absolutely unclear as to how such a law should be applied in practice. See Arthur, p. 166)

James Rachels likewise uses a reductio argument to criticize passive euthanasia.

Our premise is that in some cases, it is better to let patients die, rather than use every means possible to keep them alive.

But consider where this premise leads - in the case of Down's syndrome children:

If a Down's syndrome child is born with an intestinal blockage - one which will quickly prove fatal unless operated on (a routine procedure) - in the name of passive euthanasia, we will let the child die.

If a Down's syndrome child is born without an intestinal blockage - our commitment to passive euthanasia means that we will let the child live.

Rachels points out that "the matter of life and death is decided on irrelevant grounds" (Arthur, 173, emphasis added) - i.e., whether or not the child is born with an intestinal blockage. (Irrelevance is another, rather broad category of fallacy.) At the same time, he's showing here that the premise of passive euthanasia leads to two contradictory conclusions - thus throwing that premise into question.