Before Gov. John Kitzhaber of Oregon established a moratorium on his state’s death penalty last year, Christian Longo, a death row inmate, started a campaign to allow the condemned to donate their organs. Mr. Longo argued that a new execution protocol that many states — including Texas — have adopted leaves inmates’ organs viable for transplantation. “While I can potentially help in saving one life with a kidney donation now, one preplanned execution can additionally save from 6 to 10 more lives,” Mr. Longo wrote in a plea that Oregon officials denied. No state allows death row inmates to donate their organs. Although Texas recently abando...