An aggrieved party may obtain a review of any final judgment of the superior court under this chapter by the Court of Appeals or the Supreme Court, as provided by law. In contested cases involving a license to practice medicine or a license to practice dentistry in this state, the filing of an application for appeal or a notice of appeal shall not by itself stay enforcement of the agency decision. In such cases, the superior court which considered the petition for judicial review or the Court of Appeals or the Supreme Court may order a stay only if such court makes a finding that the public health, safety, and welfare will not be harmed by the issuance of the stay.