Can I appeal if my application for medical certification is denied?
Yes. There are many avenues of appeal--but you must do some serious soul searching and by honest with yourself. There are many medical conditions, some temporary and some premanent, that make it unsafe to pilot an aircraft. If you still disagree with your AME, then Part 67 of the Federal Aviation Regulations outlines the appeal process for applicants who are denied medical certification.
In a nutshell, the initial appeal would be to the Federal Air Surgeon (through our Oklahoma City office) to request an authorization for the special issuance of a medical certificate. This might result in a medical certificate that is time-limited, contingent upon the successful completion of addition medical testing, or otherwise restricted. If this request is not successful, then an appeal to the National Transportation Safety Board (NTSB) could be made. If the NTSB concurs with the FAA's denial action, you could then request a hearing in Federal District Court and ultimately the Supreme Court. Section 67.409 under 14 CFR sets forth the appeal process within the FAA for applicants who are denied medical certification.