I am looking for a totally online organic chemistry course.  It cannot
be an elementary or essentials course.  I do not need an accompanying
laboratory credit.  Is this available from any university??  This
course should be a prerequisite for medical, dental, or pharmacy
school.

Your question implies that you need to receive official "credit" for
this course, is that correct?  If not, and you are simply trying to
learn organic chemistry on your own, you might take a look at MIT's
OpenCourseWare materials at
http://ocw.mit.edu/OcwWeb/Chemistry/index.htm.

Athabaska University (http://www.athabascau.ca/) in Canada offers a
variety of distance learning options, including a 2-semester course in
Organic Chemistry (http://www.athabascau.ca/course/ug_subject/list_cd.php#chem).
 You can register, pay, and get credit for these courses.

About.com lists a number of additional online chemistry courses at
http://chemistry.about.com/od/onlinecourses/

Yes, I do need to earn official credit for the course.  I also found
the course at Athabaska, but I was unsure if this Canadian credit
would be credible in the US.

I recently answered a similar question. The schools listed there might
have what you're looking for.

http://answers.google.com/answers/threadview?id=506911
Q: Online course in general chemistry from accredited university