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Charlie Sykes, a popular radio host in Milwaukee, welcomed Donald J. Trump to his state in a phone interview on his program on Monday, explaining that “here in Wisconsin we value things like civility, decency and actual conservative principles.”

And then, for Mr. Trump, the 17-minute interview went downhill from there.

Mr. Sykes, an outspoken conservative and opponent of Mr. Trump, began by pressing the Manhattan businessman to apologize to Heidi Cruz, the wife of Senator Ted Cruz of Texas, for resending a Twitter post juxtaposing an unflattering picture of Mrs. Cruz with a shot of his wife, Melania Trump, a former model.

The spat began after an anti-Trump “super PAC” produced an ad geared at Utah voters featuring an image of Mrs. Trump, then Melania Knauss, posing nude from a January 2000 GQ magazine pictorial.

On Mr. Sykes’s program, Mr. Trump defended the photo of his wife — calling it “an artsy picture” — but blamed Mr. Cruz for starting the feud, even though the super PAC that produced the Texas ad is not connected to the Texas senator.

“I expect that from a 12-year-old bully on the playground, not somebody who wants the office held by Abraham Lincoln,” Mr. Sykes said.

Later, when Mr. Trump again complained, incorrectly, that Mr. Cruz “started it,” the radio host interjected: “Remember, we’re not on a playground. We’re running for president of the United States.”

Mr. Trump said he was unaware the Twitter photo of Mrs. Cruz that he resent was unflattering. “I didn’t even know it was necessarily a very bad picture of her versus Melania,” he said.

Later, pressed on the gender gap he faces with female voters, as well as disparaging comments he has made about women — including Megyn Kelly of Fox News and Carly Fiorina, his former Republican rival — Mr. Trump defended himself.

“Women are just going to have to see what I’ve done,” he said. “I’ve hired tremendous numbers of women. Women are in the highest executive positions.”

He added he’s “not a fan of Megyn Kelly” and he’d be making the exact same criticisms if she were a man.

“I don’t think there should be a double standard if somebody’s a woman,” Mr. Trump said. “People have said, you know, in terms of breaking the glass ceiling, that I was a leader in it.”

Mr. Trump, who at moments seemed caught off guard by the combative tone of the interview, had gone on the program without realizing that Mr. Sykes is a Cruz supporter who has repeatedly criticized the billionaire and said he could never support him.

“Mr. Trump, before you called into my show, did you know that I’m a hashtag ‘NeverTrump’ guy?” Mr. Sykes asked near the end of the back-and-forth, referring to the #NeverTrump movement of Republicans who are trying to stop Mr. Trump from winning the party’s nomination.

“No, I didn’t know that,” Mr. Trump said. “But I assume you’re also an intelligent guy. I know you’re an intelligent guy, and you understand what’s going on. Our country is getting ripped off left and right by every country we do business with, even in NATO.”

Mr. Sykes concluded his interview by worrying he had not indoctrinated Mr. Trump into the Wisconsin way of life.

“Obviously I failed in my effort to introduce you to Wisconsin and our tradition of civility and decency by getting an apology from you for Heidi Cruz,” he said.

But, Mr. Sykes added, he had some “free advice” for Mr. Trump, eight days before his state’s primary on April 5: “If you stand up there and you say, ‘Folks let me just say this: I’m running for president, Ted Cruz is running for president, let’s leave our wives out of it. Both of us have married intelligent, beautiful women and from now on we are not going to be talking about one another’s wives,’ people in Wisconsin will love that, Mr. Trump,” he said.