Not sure about the calendar stuff. I don't think you need links so much as you need to generating .ics files (for Outlook) or .cal? files (for Macs) and attach them to the emails.
You also need to make sure that you are complying with medical privacy laws (HIPAA here in the United States). Under HIPAA, I believe you cannot communicate medical information (including appointments) with patients without their prior signed consent to receive information that way.