The furor underscored the continuing troubles for the White House since Mr. Trump first accused Mr. Obama of tapping his phones, an allegation refuted by intelligence agencies as well as Republican and Democratic officials. Even as Mr. Trump refused to back down, fellow Republicans appeared increasingly irritated by what they see as a distraction from their policy goals.

Representative Tom Cole of Oklahoma said on Friday that Mr. Trump had not proved his case and should apologize to Mr. Obama. “Frankly, unless you can produce some pretty compelling truth, I think President Obama is owed an apology,” Mr. Cole told reporters. “If he didn’t do it, we shouldn’t be reckless in accusations that he did.”

The conspiracy theorizing also tested what is often called the special relationship between the United States and Britain. American intelligence agencies enjoy a closer collaboration with their British counterparts than any other in the world. GCHQ was the first agency to warn the United States government that Russia was hacking Democratic Party emails during the presidential campaign.

Foreign policy analysts expressed astonishment that Mr. Trump would so cavalierly endanger that partnership. “It illustrates the extent to which the White House really doesn’t care what damage they do to crucial relationships in order to avoid admitting their dishonesty,” said Kori Schake, a former national security aide to President George W. Bush now at Stanford University’s Hoover Institution. “America’s allies are having to protect themselves against being tarred with the White House’s mendacity.”

Eric S. Edelman, an under secretary of defense under Mr. Bush, has written about the stresses between the United States and Britain in recent years. “I hope that this latest episode doesn’t drive a stake through the heart of the strongest remaining element of Anglo-American partnership,” he said.

Julianne Smith, who was a deputy national security adviser to Vice President Joseph R. Biden Jr., said Mr. Trump did not appear to realize how much American intelligence agencies depend on Britain in dealing with threats around the world. “He will probably live to see the day when he will regret firing off such an egregious insult to Britain and then failing to apologize for it,” she said.

The issue clearly touched a nerve at GCHQ, which usually refuses to comment on intelligence matters. Its vehement response surprised British officials and analysts. Dominic Grieve, the intelligence committee chairman in Parliament, pointed to elaborate safeguards that prevent spying on the United States and require “a valid national security purpose” for any monitoring. “It is inconceivable that those legal requirements could be met in the circumstances described,” he said.