YANGON, Myanmar — Thailand’s aggressive campaign to protect its ailing king from criticism escalated into a diplomatic feud on Wednesday when the Thai police said they were investigating the United States ambassador to the country in connection with a royal insult.

The police said that the ambassador, Glyn T. Davies, had diplomatic immunity and could not be charged for such a crime. But the authorities are obliged by law to investigate a complaint lodged last week, said the deputy spokesman of the national police, Maj. Gen. Songpon Wattanachai.

“We have to accept the complaint and proceed with it,” he said.

General Songpon said that if comments by the ambassador were judged to be insulting to the monarchy, the government might proceed with some kind of reprimand, although he was unsure of the form it might take. “We may have to issue a warning to him,” the general said.

In a speech last month, Mr. Davies criticized the long prison sentences handed to some of those found guilty of criticizing King Bhumibol Adulyadej. The police have requested a copy of the speech and will have it translated by a “reliable institution,” General Songpon said.