The court instead directed a lower court to handle the case.
Mill's attorneys contend the judge who sentenced the musician refuses to respond to their post-sentencing appeals.
Mill was sentenced this month to two to four years in prison for violating probation on a roughly decade-old gun and drug case.
The sentence has been criticized as extreme, but Common Pleas Court Genece Brinkley has said Mill ignored the rules of his probation.
Mill's legal team was asking Brinkley to step down from the case, saying she has made the case inappropriately personal.
The motion filed Monday contended Mill can't appeal until Brinkley rules on his post-sentence motions, and she "refuses to do so."
Mill's case has attracted attention through rallies, billboards and buses in Philadelphia carrying the message "Stand With Meek Mill."
The Rev. Al Sharpton met with Mill on Monday.
"We had a very blunt talk," said Sharpton. "He's a bright young man. He's committed and he really said 'Reverend, if nothing else, let people understand what this system does.'"
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