Mr. Trump stands at the center of that debate, a pugilist and partisan who rarely describes himself as the leader of all Americans. The past three presidents all came to office promising in different ways to be, as Mr. Bush put it, “a uniter, not a divider,” even if none of them fully realized the ambition in polarizing times. But that has never really been Mr. Trump’s stated goal, much less his approach to the office.

A reality television star who praised a congressman for body-slamming a reporter and urged supporters to “knock the crap” out of protesters, he has connected with his admirers through a combative style that feels invigorating to those who resent the longtime establishment Mr. Trump promises to take on. While the president may have a point about the larger tenor of discourse, which precedes and exceeds him, Mr. Trump does not subscribe to the traditional belief that a president should meet a higher standard to set an example for everyone else.

Even as the authorities were finding more pipe bombs targeting his critics on Thursday morning, Mr. Trump turned his fire at the news media. “A very big part of the Anger we see today in our society is caused by the purposely false and inaccurate reporting of the Mainstream Media that I refer to as Fake News,” he wrote on Twitter. “It has gotten so bad and hateful that it is beyond description. Mainstream Media must clean up its act, FAST!”

Mr. Trump did not mention that CNN, the news network that he has long assailed and called “fake news,” was among the targets of a pipe bomb, along with former President Barack Obama, former Secretary of State Hillary Clinton and, as of Thursday, former Vice President Joseph R. Biden Jr. and the actor Robert De Niro. At his rallies, Mr. Trump singles out CNN for opprobrium, prompting the crowd to chant “CNN sucks! CNN sucks!”

John O. Brennan, the former C.I.A. director who has become one of Mr. Trump’s biggest critics and was one of the intended targets of a pipe bomb, said on Thursday that the president was trying to deflect his own responsibility for the ugly environment.