Can I appeal if my application for medical certification is denied?

Yes. Section 67.409 under 14 CFR sets forth the appeal process within the FAA for applicants who are denied medical certification.

Within 30 days after the date of the denial, you may apply for reconsideration to:
Federal Air Surgeon
Attention: Manager, Aerospace Medical Certification Division, AAM-300
Federal Aviation Administration
P.O. Box 25082
Oklahoma City, Oklahoma 73126