WASHINGTON — When last they saw each other six weeks ago after the ceremonial passing of power, President Trump and former President Barack Obama parted with smiles and handshakes. But it did not take long for the surface bonhomie to degenerate into a fierce and public clash unlike any other in modern times.

While Mr. Obama has remained quiet for the most part, some of his closest loyalists moved into opposition mode, leading what some only half-jokingly call “the resistance.” Mr. Trump, convinced that Obama holdovers still in government are trying to sabotage his presidency, took the conflict nuclear over the weekend by accusing his predecessor of bugging his telephones last year.

Mr. Trump provided no proof, and the charge was quickly dismissed by intelligence veterans and, indirectly, the F.B.I., but that did not make it any less sensational or any less historic. Never in recent generations has the natural friction between current and past presidents spilled over into such a public spectacle. If sustained, it could fray the institution of the presidency, further erode the public’s already low confidence in the nation’s leadership and leave both allies and enemies with the impression of an America at war with itself.

Perhaps it should be no surprise that it came to this between the 44th and 45th presidents. During last year’s campaign, Mr. Obama called Mr. Trump a “con artist and a know-nothing” who could not be trusted with the nuclear codes. Mr. Trump called Mr. Obama “the worst president in the history of the United States” after spending years questioning whether he had been born in the United States. They put that rancor aside for a cordial meeting after the election, but that barely veiled the chasm between them in terms of personality, politics and policy.