In the United States, must state courts follow rulings by federal courts of appeals?
Widely accepted answer: no, state courts are not bound by circuit precedent
The near-consensus is that state courts need not follow the rulings by federal courts of appeal.1 State courts are coordinate and coequal with the lower federal courts on matters of federal law.2 Only United States Supreme Court precedent binds state courts on matters of federal law. And state courts are supreme, even relative to the Supreme Court of the United States, on matters of state law.3 See this summary from the Georgetown Law Writing Centre (pp. 4-5).
Amanda Frost says that the "conventional wisdom" is that "lower federal court precedent cannot bind state courts" ("Inferiority Complex: Should State Courts Follow Lower Federal Court Precedent on the Meaning of Federal Law?" (2015)):
Most state courts assert that they are free to reach their own conclusions about the meaning of federal law, even when doing so creates a conflict with the federal court of appeals presiding over the geographic region in which they sit. Several federal circuits have conceded that their decisions are not binding on state courts... A number of federal courts scholars have declared that state courts need not follow lower federal court precedent because state courts are "coordinate" with lower federal courts and not "subordinate" to them.
Frost presents several state-circuit splits. For example (as of 2015):
In 2000, the Texas Court of Criminal Appeals held that the Fifth Amendment requires only that law enforcement inform a suspect that he has a right to counsel prior to interrogation, without specifying that counsel may be present during the interrogation. That decision is in direct conflict with the 1968 decision by the U.S. Court of Appeals for the Fifth Circuit holding that a suspect must be informed that he has a right to counsel before and during interrogation. Accordingly, the standard for Mirandizing a suspect in the state of Texas varies depending on whether the case would be tried in state or federal court.
Frost goes on to argue for a change: that "under some circumstances," state courts should be mandated to "follow the precedent of the federal court of appeals for the geographic region in which that state is located" (at p. 62).
The relationship between the federal circuits and state courts is also summarized by Wayne A. Logan in "A House Divided: When State and Lower Federal Courts Disagree on Federal Constitutional Rights" (2014):
... state courts have long shared a concurrent obligation with lower federal courts to interpret the U.S. Constitution and protect the rights contained in it. Cricially important as well, state courts do so independently of their federal counterparts. They need not defer to the constitutional positions adopted by federal circuit courts, including those in which they are geographically situated, which lack direct appellate review authority over them. As a consequence, on all issues other than the comparatively narrow range of questions expressly addressed by the Supreme Court, state and lower federal courts are free to disagree...
One of the citations within the above passage quotes from the Fourth Circuit:
Though state courts may for policy reasons follow the decisions of the Court of Appeals whose circuit includes their state, they are not obliged to do so.
But the Ninth Circuit says state courts are bound; California and the Supreme Court disagree
However, the Court of Appeals for the Ninth Circuit is somewhat obstinate in its position that the state courts within its boundaries are bound by its precedent. California state courts disagree. And apparently, so does the United States Supreme Court. It commented on the Ninth Circuit's position in Arizonans for Official English v. Arizona, 520 U.S. 43 (1997), calling the Ninth Circuit's position "remarkable." See footnote 11:
The Court of Appeals questioned the wisdom of the view expressed "in the academic literature," "by some state courts," and by "several individual justices" that state courts are "coordinate and coequal with the lower federal courts on matters of federal law." ... But c.f. ASARCO Inc. v. Kadish, 490 U.S. 605, 617 (1989) ("state courts . . . possess the authority, absent a provision for exclusive federal jurisdiction, to render binding judicial decisions on their own interpretations of federal law"); Lockhart v. Fretwell, 506 U.S> 364, 375-376 (1993) (Thomas J. concurring) (Supremacy Clause does not require state courts to follow rulings by federal courts of appeals on questions of federal law).
1. Do not confuse state courts (which are entire judicial systems within each state) with federal district courts (which are the trial-level courts of the federal judiciary, necessarily distributed around the states within a circuit). Federal district courts, over which circuit courts of appeal have direct appellate review, are bound by their circuit precedent.
2. In this context, "lower federal courts" means federal courts other than the Supreme Court of the United States. Circuit precedent is set by their Courts of Appeals.
3. But see Martin v. Hunter's Lessee (1816); Michigan v. Long, 463 U.S. 1032 (1983). Where the validity of a state law or the availability of a remedy for state action depends on the interpretation of federal law, including interpretation of the US Constitution, this is a question of federal law, reviewable by the Supreme Court of the United States.