According to Emily Post, a letter to a Supreme Court judge is addressed using "The Chief Justice" without a name or "Justice" with the last name. The full name of the court is under the name, followed by the address.
According to the Law Dictionary, correspondence to a judge in a lower court is addressed with "The Honorable" followed by the judge's full name. The court name and address follow. Emily Post indicates that the greeting in a letter to a Supreme Court judge uses the judge's title, such as "Chief Justice" with no further name or "Justice" with the last name. "Judge" with the last name is used for those who sit on lower courts.