The Fox News host Sean Hannity has been very upset.

Mr. Hannity is often very upset, at things like the past friendships of President Barack Obama; the suspected political leanings of his former colleague Megyn Kelly; the resignation of Michael T. Flynn, President Trump’s former national security adviser; and the tweets of media reporters from rival news organizations.

But the primary target of his anger in recent days has been Ted Koppel, a veteran news anchor and contributor to “CBS Sunday Morning,” which last weekend broadcast a clip from an interview with Mr. Hannity that the Fox star did not like at all.

The interview was conducted by Mr. Koppel roughly three weeks ago, Mr. Hannity said on his show Monday night, and had been presented to him as an opportunity to discuss the political polarization in American life. But only a minute or so of the full interview was used. And it looks as though it may have been the part that Mr. Hannity found the most insulting.

The night before the segment aired, CBS News posted online a part of the interview in which Mr. Hannity described his upbringing, which he said shaped his views. But in the televised footage, he was shown complaining about socialism, liberalism and “angry snowflakes,” right-wing slang for liberals that conservatives see as easily upset.

Then he lead with his chin: Mr. Hannity asked if Mr. Koppel, an elder statesman of broadcast news who hosted the ABC show “Nightline” for 25 years, thought he and his show were “bad for America.” Mr. Koppel responded, “Yeah.”

“Really?” Mr. Hannity responded, his voice rising an octave or two. “That’s sad, Ted. That’s sad.”

Why did Mr. Koppel think Mr. Hannity’s show was bad for America?

“You have attracted people who have determined that ideology is more important than facts,” Mr. Koppel said. He also told Mr. Hannity, an enthusiastic on-air supporter of President Trump, “You are very good at what you do.”

What Mr. Hannity has done for the last two nights is use his show to put his anger on full display.

He has directly addressed Mr. Koppel (“Ted, you show in this clip, you are not a journalist,” he said on Monday’s show.) He has denounced the segment as “edited fake news.” He has compared Mr. Koppel to Barack Obama and Hillary Clinton — perhaps the worst insult a Fox News host can throw at you.

He has also demanded that CBS release the full tape of his interview, which he called “a very good, substantive give-and-take.”

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Bill O’Reilly, another Fox host, and Brit Hume, a political analyst at the network, repeated that call on Monday night.

So did the conservative commentator Michelle Malkin, who called the 77-year-old Mr. Koppel “ancient” and a “walking dead decrepit media elitist.” On Tuesday’s broadcast, the conservative radio host Laura Ingraham told Mr. Hannity that he was Mr. Koppel’s “victim” and said CBS treated him unfairly as a way to protect “the worldview of the left.”

“They want to delegitimize you,” she said.

Mr. Koppel has not responded to Mr. Hannity. A spokesman for CBS News said the network would not respond, either, and also would not release the unedited footage of the interview, which was part of a 10-minute segment that was not a profile devoted to Mr. Hannity.

Mr. Hannity has turned the slight into a dispute over the intelligence of “the American people,” which he accused Mr. Koppel of underestimating. He said people were smart enough to know the difference between what he does and a news broadcast.

So what does he do? On Monday, Mr. Hannity described himself as “a talk-show host” and “an advocacy journalist” who covers stories — like President Obama’s ties to “black liberation theology,” he said in one example — that CBS would not touch.

But he also used the word “news” to describe the service he provides his viewers, asking Mr. Koppel, “How can I be bad for America when I am offering the American people news and information nightly that your own network will not touch because they have an agenda?”

Mr. Hannity has never hidden his own conservative beliefs. Immediately before Mr. Koppel’s rebuke, he was describing a right-wing political vision so expansive that he went out of his way to be clear that he did not mean to foment revolution.

“Honestly, I think liberalism has to be defeated. Socialism must be defeated — in a political sense,” he said. “This is not a — we don’t want a revolution in this country.”

One of the prime engines of left-wing disloyalty was the media, he said, which he accused of being “out to destroy the president.” He now says Mr. Koppel is doing the same thing to him.

“That’s the difference, Ted, between me and you,” he said on Monday night. “I’m honest with my audience. I don’t pretend that I’m fair and balanced and objective. You do.”