President Trump complained Wednesday that he was never thanked for allowing Sen. John McCain’s funeral celebration, escalating a feud with the McCain family and drawing the ire of senators on Capitol Hill who told the president he was embarrassing himself.

Mr. Trump, during a speech to workers at an Army tank plant in Ohio, took a detour in his remarks to blame the press for picking at years of scar tissue between the two men which has lasted even beyond McCain’s death in August.

The president said he has “certain reasons” for disliking the former senator, including his votes to sink an Obamacare-repeal bill and his role in helping ensure the salacious anti-Trump dossier compiled by Christopher Steele made its way to the FBI in 2016.

But Mr. Trump added new complaints, too, saying he didn’t get the appreciation he deserved after the week-long commemoration of McCain last summer.

“I endorsed him at his request and I gave him the kind of funeral that he wanted, which as president I had to approve,” Mr. Trump said. “I didn’t get thank-you’d — that’s OK. We sent him on the way. But I wasn’t a fan of John McCain. “

The remarks were the most vehement in what’s been days of gripes about McCain, and the criticism of a revered dead colleague had begun to sour even staunch Trump allies.

“I think the president’s comments about Senator McCain hurt him more than they hurt the legacy of Senator McCain,” said Sen. Lindsey Graham, a Republican who has counted both men as confidantes.

But Mr. Graham also said he was struck by the hosannas for McCain from people who used to attack the late senator, particularly in 2008, when McCain was the Republican presidential nominee.

“A lot of people are coming to John’s defense now that called him crazy and a warmonger,” Mr. Graham said.

He didn’t name names.

Sen. Charles E. Schumer, Democrats’ floor leader, leaped to McCain’s defense Wednesday, tweeting that he will pursue a resolution renaming one of the Senate’s office buildings after the late senator.

“I look forward to soon re-introducing my legislation re-naming the Senate Russell Building after American hero, Senator John McCain,” Mr. Schumer said.

He first raised the idea immediately after McCain’s death, but The Washington Times can find no resolution introduced by Mr. Schumer last year calling for the renaming.

The Democrat’s office didn’t respond to a request seeking clarification.

Other Trump opponents also rushed to capitalize.

Former Ohio Gov. John Kasich sent out a fundraising plea to supporters blasting Mr. Trump over his comments.

“When we don’t speak out about these things, we become numb to them and we simply can’t let that happen,” wrote Mr. Kasich, who lost the GOP nomination to Mr. Trump in 2016 and has not publicly ruled out a repeat attempt in 2020.

Sen. Johnny Isakson, Georgia Republican, said Mr. Trump was showing a lack of respect for McCain’s service to the country.

“I don’t care if he’s president of United States, owns all the real estate in New York, or is building the greatest immigration system in the world, nothing is more important than the integrity of the country and those who fought and risked their lives for all of us,” Mr. Isakson told The Bulwark, a website and podcast.

Mr. Trump insulted McCain during the 2016 election when the novice candidate seemed to mock the senator’s service record as a Navy aviator, including getting shot down and surviving years in a prisoner of war camp during the Vietnam war.

They forged some common ground later in the campaign, but once the election was over, they found themselves at odds again.

McCain cast a deciding vote to sink the GOP’s effort to repeal Obamacare in summer 2017 and led criticism of Mr. Trump’s attempts to foster a diplomatic relationship with Russian President Vladimir Putin.

Even death did not cool their feud. Mr. Trump was left out of McCain’s funeral proceedings, and the late senator’s daughter, Meghan McCain, has used her platform as a talk show figure to criticize the president.

The feud was renewed last week by reports, based on unsealed court documents, that McCain had deployed an aide to be a conduit for the salacious and unverified anti-Trump dossier that rocked the Trump transition, suggesting unproved contacts with Russian figures.

Mr. Trump took to Twitter over the weekend to complain about McCain, prompting Ms. McCain to fire back that her father was loved — and Mr. Trump isn’t. Reporters then prodded Mr. Trump for a reply and he complied, doubling down on his animosity.

On Wednesday, speaking at the tank plant in Ohio, Mr. Trump said he figured it was time to settle things.

“John McCain received a fake and phony dossier,” Mr. Trump told the crowd. “It was paid for by crooked Hillary Clinton. And John McCain got it. And what did he do? He didn’t call me. He turned it over to the FBI, hoping to put me in jeopardy. That’s not the nicest thing to do. I’m a very loyal person.”

Mr. Trump also blamed McCain for getting America involved in the wars in Iraq and Afghanistan after the Sept. 11 terrorist attacks.

“We’re in a war in the Middle East that McCain pushed so hard,” Mr. Trump said. “He was calling President [George W.] Bush all the time — ‘Get into the Middle East, get into the Middle East.’ So now we’re in that war for $7 trillion, thousands and thousands of our people have been killed, millions of people overall, and frankly, we’re straightening it out now, but it’s been a disaster for our country. We’ve spent tremendous wealth, and tremendous lives in that war. And what do we have? It’s worse than it was 19 years ago. I call them the endless wars. So John McCain loved it.”

• Gabriella Muñoz and Bailey Vogt contributed to this article.

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