The only state courts eligible to have cases reviewed by the US Supreme Court are the state supreme courts or court of appeals immediately under the state supreme court (if that court declines review), because a case must exhaust all appellate options before any party can petition the US Supreme Court for a writ of certiorari.
State supreme courts can avoid having their cases reviewed by never accepting a case that involves questions of federal or constitutional law. Barring that, they could pay careful attention to case law and not make a ruling in conflict with an established precedent.
Barring that, they can only hope whoever loses the state supreme court appeal doesn't petition the US Supreme Court for a writ of certiorari or, if they do, hope that the case isn't sufficiently compelling that it attracts four of the Justices' interest.
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supreme court
The US Supreme Court set a standard on the power of judicial review. This meant, that the Court could review governmental actions without a lawsuit to be settled. The Court, on its own volition had the right to review issues that pertained to the US Constitution. In effect, the Court already had the ability or practice of " judicial review". It did not "gain it", it simply used the power the US Constitution gave to the Court.
A supreme court is the highest court within the hierarchy of many legal jurisdictions. Other descriptions for such courts include court of last resort, instance court, judgment court, apex court, and highest court of appeal. Broadly speaking, the decisions of a supreme court are not subject to further review by any other court.
the supreme court
Judicial Review