President Trump’s former Chief of Staff John F. Kelly said late Wednesday the National Security Council staffer ousted from the White House was just doing his job when he raised concerns about the president’s July phone call with the Ukrainian president.

Mr. Kelly defended Lt. Col Alexander Vindman, who was escorted from the White House last week, during a speech late Wednesday to college students in New Jersey.

“He did exactly what we teach them to do from cradle to grave,” Mr. Kelly said at Drew University in Morristown, according to The Atlantic. “He went and told his boss what he just heard.”

Mr. Trump’s decision to transfer the decorated veteran to the Defense Department was seen as the first round of retribution in the wake of the president’s acquittal on impeachment articles related to Ukraine.

Mr. Kelly, a retired U.S. Marine Corps general, served as secretary of Homeland Security and then as Mr. Trump’s second chief of staff from mid-2017 to January 2019. He was often portrayed as a check on Mr. Trump’s wilder impulses.

Mr. Kelly spoke about Mr. Trump in unusually critical terms during his New Jersey appearance, questioning his approach on everything from immigration to the media to North Korea.

Mr. Trump hit back on Twitter on Thursday, saying Mr. Kelly was not cut out to be a chief of staff and, like other former aides, “can’t keep his mouth shut.”

Mr. Kelly said Lt. Col. Vindman had to object to Mr. Trump’s July 25 call with Ukrainian President Volodymyr Zelensky because the president appeared to be reversing long-standing policy toward the country and conditioning U.S. aid on investigations into former Vice President Joseph R. Biden, a 2020 foe.

Mr. Kelly said it amounted to an “illegal order” that other federal agencies should know about.

“We teach them, ‘Don’t follow an illegal order. And if you’re ever given one, you’ll raise it to whoever gives it to you that this is an illegal order, and then tell your boss.’” Mr. Kelly said in The Atlantic’s account.

The retired general said Mr. Trump’s decision to intervene in the Edward Gallagher case, after the Navy SEAL’s conviction for violating military law, was “exactly the wrong thing to do,” and said North Korean leader Kim Jong-un has been “playing” Mr. Trump.

“He will never give his nuclear weapons up. I never did think Kim would do anything other than play us for a while, and he did that, fairly effectively,” Mr. Kelly said, according to an account in The Daily Record, a paper serving Morris County, New Jersey.

Mr. Kelly said he didn’t like Mr. Trump’s travel ban targeting Muslims and his decision to dub the media the “enemy of the state.”

He said he doesn’t think the border wall needs to extend “from sea to shining sea,” the Atlantic reported, and said many migrants are good people looking for opportunity, and not criminal or “rapists” as Mr. Trump famously put it in 2015.

Mr. Kelly replaced former Republican National Committee Chairman Reince Priebus as chief of staff during Mr. Trump’s first year in office. He said his wife encouraged him to take on the role, saying the president could use a guy like him.

He later gave way to acting Chief of Staff Mick Mulvaney, who has been more content to let Mr. Trump act as he sees fit than intervene or control who sees the president.

“When I terminated John Kelly, which I couldn’t do fast enough, he knew full well that he was way over his head,” Mr. Trump tweeted Thursday. “Being Chief of Staff just wasn’t for him. He came in with a bang, went out with a whimper, but like so many X’s, he misses the action & just can’t keep his mouth shut, which he actually has a military and legal obligation to do.”

“His incredible wife, Karen, who I have a lot of respect for, once pulled me aside & said strongly that ‘John respects you greatly. When we are no longer here, he will only speak well of you,’ ” Mr. Trump wrote. “Wrong!”

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