The case you're referring to is Herrera v. Collins, 506 U.S. 390 (1993). The relevant excerpt is in Justice Antonin Scalia's concurrence [1]:
> There is no basis in text, tradition, or even in contemporary practice (if that were enough), for finding in the Constitution a right to demand judicial consideration of newly discovered evidence of innocence brought forward after conviction.
...
> With any luck, we shall avoid ever having to face this embarrassing question again, since it is improbable that evidence of innocence as convincing as today's opinion requires would fail to produce an executive pardon.
Note: In the excerpt above from source [1], Snopes removed a page break and inserted a comma (mistake?). Keep that in mind when using ctrl-F in source [2].
The Supreme Court's ruling (6-3) was that someone who was already convicted for a crime but later makes a claim of new evidence of innocence is not entitled for a new judicial trial [2]. In simple terms, there's no guarantee to a new trial after the fact, and the burden on the courts would be too much anyway. Scalia's concurrence went further and asserted that executing someone who was convicted in a procedurally proper trial but was actually innocent is not "cruel and unusual punishment" (by the standards of the past 200+ years of US history, which supposedly involved plenty of executions of innocent people convicted in proper trials) and is not a violation of due process.
The majority (including Scalia) weren't in favor of executing an innocent person, they were just arguing that the convict making the new claim of innocence can't count on the judicial system. The majority, including Scalia, points to the only option being seeking executive clemency/pardon, i.e. request the governor/president to evaluate the new evidence of supposed innocence and hope that the governor/president agrees - or at least reduces the sentence [2]. A request for clemency is outside of the scope of due process.
> Herrera is not left without a forum to raise his actual innocence claim. He may file a request for clemency under Texas law, which contains specific guidelines for pardons on the ground of innocence. History shows that executive clemency is the traditional "fail safe" remedy
[page break]
> for claims of innocence based on new evidence, discovered too late in the day to file a new trial motion.
From the convict's perspective, "no new trial, but you can ask for a pardon" is cold comfort, but not necessarily a dead end.
[1] https://www.snopes.com/fact-check/scalia-death-penalty-quote...
[2] https://supreme.justia.com/cases/federal/us/506/390/ https://supreme.justia.com/cases/federal/us/506/390/case.pdf