[The 9th and SCOTUS denied his request to appeal to them. They did not rule on the legality of Bolton denying Arpio's request. ]
Are you new to the common law system?  Refusing an appeal _is_ a ruling on the legality of the request.
[Bolton violated the law by denying him a jury trial.]
What is the law in question?  That an accused person is entitled to a jury trial.  This is without limitation?  The Supreme Court determined various limitations when it comes to jury trial.  For example, jury trials with six people empaneled must be unanimous but not twelve.  But to the actual case:
No jury trial is required, however, when the maximum sentence is six months in jail, a fine not to exceed $1,000, a 90-day driver's license suspension, and attendance at an alcohol abuse education course.  Blanton v. City of North Las Vegas, 489 U.S. 538 (1989)
How long was the maximum sentence Arpaio faced?  Six months.