[Mr. Lucky]

Justice Roberts has a vote in determining whether to accept an appeal from a state court; he has no other authority over them.

[Political Junkie Too]

Each Justice sits over one or more district courts. Those courts hear the appeals from the states in their respective districts.

Roberts could influence the other Justices to let the districts (and by extension, the states) what the "sense" of the Supreme Court is, and that they won't tolerate cases coming from the states that go against the SCOTUS rulings.

It's a whisper campaign, court style.

-PJ