“I told him that although the phrase ‘fake news’ is untrue and harmful, I am far more concerned about his labeling journalists ‘the enemy of the people,’” Mr. Sulzberger continued. “I warned that this inflammatory language is contributing to a rise in threats against journalists and will lead to violence.”

This is particularly true overseas, Mr. Sulzberger said, where governments are using Mr. Trump’s words as a pretext to crack down on journalists. He said he warned the president that his attacks were “putting lives at risk” and “undermining the democratic ideals of our nation.”

Mr. Sulzberger’s lengthy, bluntly worded rebuttal was a striking rejoinder to the president by the 37-year-old publisher of a paper with which Mr. Trump has had a long, complicated relationship. And it apparently touched a nerve: The president fired off a series of angry tweets in the afternoon, accusing newspapers of being unpatriotic.

“I will not allow our great country to be sold out by anti-Trump haters in the dying newspaper industry,” he wrote. “The failing New York Times and the Amazon Washington Post do nothing but write bad stories even on very positive achievements — and they will never change!”

Mr. Trump, in his initial tweet from his golf club in Bedminster, N.J., on Sunday morning, described the meeting with Mr. Sulzberger as “very good and interesting.” But in referring to the phrase “enemy of the people,” he did not make clear that he himself began using that label about the press during his first year in office.