In a sign of how urgent the crisis is, the World Bank said in a 2014 report that 52 percent of adult men in the Polynesian kingdom of Tonga were estimated to be obese — the highest rate of 188 countries surveyed. It also said that of the seven countries worldwide with female obesity rates of at least 50 percent, four were Pacific island nations: Tonga, Samoa, Kiribati and the Federated States of Micronesia.

A study that year in the journal Diabetes Research and Clinical Practice said 10 countries and territories in the region had diabetes rates of between 19 percent and 37 percent. The rate in Vanuatu, which has a population of about 250,000, was nearly 24 percent, the study said, and economic development there was closely linked with consumption of animal proteins and simple carbohydrates.

By contrast, the diabetes rate in the United States in 2014 was 9.3 percent.

“It is so wrong what is being done to exploit these nations by providing a food supply that is not, in the long term, better for health,” said Elaine Rush, a professor of nutrition at the Auckland University of Technology in New Zealand who has studied health problems in the Pacific islands. She described the effect that the health crisis was having on families there as “decimating.”

Experts said in interviews that there was a precedent in the region for policies to restrict imports of unhealthy drinks and other products.

More than half of the 20 Pacific island countries and territories monitored by the World Health Organization, including Vanuatu, have taxes for sugar-sweetened beverages, said Wendy Snowdon, a W.H.O. official based in Suva, the capital of Fiji. Some communities in the region have banned tobacco, she added, and Tokelau, a territory of New Zealand, has banned imports of carbonated soft drinks.

Gerhard Sundborn, an epidemiologist at the University of Auckland, said that sugar-sweetened beverages carried an average tax of between about 7 percent and 15 percent in the Pacific islands, and that governments there typically used the taxes to discourage consumption.

Still, Dr. Snowdon said, the taxes and prohibitions on drinks in the Pacific islands — along with education, food labeling and school-nutrition programs — have not reduced the region’s overall incidence of obesity or its associated health problems.