Final answer:
The U.S. Supreme Court case Gregg v. Georgia decided that the death penalty does not inherently violate the Constitution. This 1976 decision upheld Georgia's reformed death penalty statute, signaling that death penalties could be constitutional when applied with non-arbitrary methods and criteria.
Step-by-step explanation:
The U.S. Supreme Court case that decided the death penalty itself did NOT violate the Constitution is Gregg v. Georgia, 428 U.S. 153 (1976). In this landmark decision, the Court ruled that Georgia's new death penalty statute was constitutional, thereby ending the nationwide moratorium on executions that had begun with Furman v. Georgia.
The Court's decision held that the death penalty was not a violation of the Eighth and Fourteenth Amendments as long as the method of sentencing was not arbitrary and there was a reasonable system for narrowing the class of defendants eligible for the death penalty. Prior to Gregg v. Georgia, the Court found in Furman v. Georgia that the arbitrary and inconsistent imposition of the death penalty did constitute cruel and unusual punishment.
However, Gregg and subsequent rulings provided states with guidance on how to structure their death penalty statutes in a way that would be constitutional.