Many years ago, an anthropology professor mentioned a Polynesian
society in which obesity in women was celebrated as a sign of beauty,
and bulimia and anorexia were unheard of.  However, within a year of
the introduction of American television, eating disorders sky-rocketed
and women throughout the community sought to be thin.

Can anyone tell me who these people were/are?

Thanks in advance for your answers!    

Can anyone tell me what culture or island this might have referred to?

Dear kairosea-ga, 

It would appear that it was Fiji. Anne E. Becker studied women in the
Nadroga province which did not have television until 1995.

An introduction to the topic appears on the Internet FAQ Archive.

?Even though the health focus has been on the increase in obesity and
diabetes, a different problem has occurred in Fiji. A dramatic
increase in disordered eating among teenage girls has been observed in
this nation, beginning with the introduction of television in 1995. In
1998 a researcher on Fiji reported that:
74 percent of girls reported feeling "too big or fat" at least sometimes. 
Of those who watched television at least three nights per week, 50
percent perceived themselves as too fat and 30 percent were more
likely to diet.
62 percent reported dieting in the previous month, a comparable or
higher proportion than reported in U.S. samples.?
http://www.faqs.org/nutrition/Ome-Pop/Pacific-Islanders-Diet-of.html


These are two studies which have been written on the subject.

Eating behaviours and attitudes following prolonged exposure to
television among ethnic Fijian adolescent girls.
?The purpose of this study was to evaluate the impact of the recent
introduction of Western television on disordered eating among ethnic
(indigenous) Fijian adolescent girls ? a relatively media-naïve
population in which disordered eating previously was thought to be
rare.?
http://bjp.rcpsych.org/cgi/content/full/180/6/509?ijkey=d3acdb91ca9c34c9537a9b6cdd64abae09810dcf

Changes in Prevalence of Overweight and in Body Image among Fijian
Women between 1989 and 1998.
http://www.obesityresearch.org/cgi/content/full/13/1/110

These news items refer to these papers.
http://news.bbc.co.uk/1/hi/health/347637.stm
http://www.hms.harvard.edu/news/releases/599bodyimage.html


You may also be interested in this book to which the above studies refer. 
Body, Self, and Society: The View from Fiji (New Cultural Studies
Series) (Paperback)
by Anne E. Becker 
http://www.amazon.com/gp/product/0812213971/103-4716731-1188668?v=glance&n=283155

I hope this answers your question. If it does not, or the answer is
unclear, then please ask for clarification of this research before
rating the answer. I shall respond to the clarification request as
soon as I receive it.
Thank you
answerfinder

Search strategy
polynesian island fat obesity television
://www.google.co.uk/search?hl=en&q=polynesian+island+fat+obesity+television&btnG=Google+Search&meta=

Spectacular.  Thanks very much for the thoroughness of your reply!

Dear kairosea-ga,
Thank you for the tip. Pleased I could help.
answerfinder-ga