The MD that UC Irvine (actually it went from California College of Osteopathic Medicine to California College of Medicine before it became UC Irvine - see school history website for details) awarded to DOs isn't technically an honorary degree just because it wasn't awarded honoris causa (and usually honorary degrees have the word honorary on it). It was just a worthless degree awarded (and a major reason why hospitals and other state medical boards didn't recognize it, dooming the CA DO-MD merger)
Also, it is customary to not list honorary degrees after one's name - otherwise, every celebrities/politicians will have lots of "doctorates" listed after their name. It is also customary to not call honorary doctorate degree holders "doctor". However, it is just academic custom and not a definite rule.
If the honorary degree is listed, usually it is followed by the (Hon) designation, not just (H).
In this case with Dr. Gordon, the MD(H) actually might be related to the fact that he is an MD that is board certified by the State of Arizona Board of Homeopathic Medical Examiner.
http://home.mindspring.com/~bhme/phy_dir.htm
(Don't ask me why a State board's website is located on mindspring)
Also, in the United States, the MD degree isn't awarded anymore as an honorary degree. Usually the degree of choice would be the DSc. (Doctor of Science) although at some schools, the Doctor of Science degree is still being offered as an earned degree. Other popular honorary degrees in the US include Doctor of Humane Letters, Doctor of Law, and sometimes Doctor of Music.
P.S. Some examples of people who have received honorary doctorates but society does not refer to as "doctor"
Bob Dylan (Robert Alan Zimmerman) - Honorary doctorates from Princeton University (1970) and St. Andrews University Scotland (2004)
Steven Tyler (Aerosmith) - Honorary doctorate from Berklee School of Music 2003
Mike Tyson - Honorary Doctorate of Humane Letters from Central State University Ohio in 1989 (source: Sports Illustrated, May 20, 2002, Volume 96, no. 21, Page 37)
Ted Turner (CNN AOL Time Warner) has 31 honorary doctorates