Can the US Supreme Court invoke the death penalty?

Answer 1

No. The US Supreme Court doesn't sentence defendants; the penalty phase of a trial is conducted by the original trial court.

The Supreme Court may evaluate sentencing guidelines for constitutionality, which was the basis of the Court declaring a de facto moratorium on capital punishment in 1972. In the consolidated cases of Furman v. Georgia, 408 US 238 (1972), the Supreme Court held that death penalty sentencing guidelines were inconsistent across the United States, creating an infringement on the Eighth Amendment prohibition against cruel and unusual punishment. After the states and federal government applied safeguards against arbitrariness, the Court reversed its stance in Gregg v. Georgia, 428 US 153 (1976), holding the statutory revisions made use of the death penalty constitutional again.*

They may also evaluate sentencing on an case-by-case basis, which is more common, if the question(s) the petitioner (plaintiff) raised on appeal allege the penalty somehow infringed his or her constitutional protections. The Court may decide a death penalty sentence has been misapplied, the jury was given incorrect instructions, or another matter relevant to the case was unconstitutional. In such a case, they would either vacate the sentence or reverse the judgment and remand (return) the case to a lower court for disposition. The lower court would have to resentence, grant a new trial, or release the defendant, in accordance with the justices' opinion.

Absent a new trial, a defendant can't receive a more severe penalty than the judge or jury pronounced following the original conviction.

The US Supreme Court never substitutes a new penalty of its own.

* It is important to note that 15 states and the District of Columbia no longer use capital punishment. The federal government may prohibit use of capital punishment under the Constitution, but can not mandate its use. In states that abolished the death penalty, the state constitution and laws prevail.