President Trump acknowledged his respect Monday for the late Sen. John McCain’s service to the nation and ordered U.S. flags lowered to half-staff, after complaints from veterans groups and others that the White House was snubbing the president’s frequent critic even in death.

Mr. Trump issued a proclamation Monday afternoon directing all U.S. flags to be lowered until sunset Sunday, when the 81-year-old Vietnam War hero and former presidential candidate will be buried at the U.S. Naval Academy in Annapolis, Maryland.

The White House flew its flags at full staff for most of the day after an initial show of half-staff official mourning on Sunday.

In a statement, the president paid tribute to the late Arizona Republican, with whom he clashed repeatedly since Mr. Trump announced his candidacy for the presidency in 2015.

“Despite our differences on policy and politics, I respect Sen. John McCain’s service to our country and, in his honor, have signed a proclamation to fly the flag of the United States at half-staff until the day of his interment,” Mr. Trump said.

The president, who isn’t invited to Mr. McCain’s funeral, asked Vice President Mike Pence to speak at a ceremony honoring Mr. McCain at the Capitol on Friday.

At the request of the McCain family, Mr. Trump ordered military transport of Mr. McCain’s remains from Arizona to Washington, with military pallbearers and band support, and a horse and caisson transport during the service at the Naval Academy.

Finally, the president asked White House Chief of Staff John F. Kelly, Defense Secretary James N. Mattis and White House National Security Adviser John R. Bolton to represent the administration at the services.

At a White House dinner celebrating evangelical leadership Monday night, Mr. Trump said, “Our hearts and prayers are going to the family of Sen. John McCain.”

“We very much appreciate everything that Sen. McCain has done for our country,” the president said.

In making these gestures for Mr. McCain and his family, Mr. Trump seemed willing, if grudgingly, to put in the past his famous and long-running feud with Mr. McCain. It began in 2015 when the senator blamed Mr. Trump for inciting anti-immigration “crazies” at an Arizona rally and Mr. Trump hit back that Mr. McCain was a war hero only because “he was captured.”

“They had a tense relationship on both sides,” said Christopher Ruddy, a friend of Mr. Trump who also counted himself as a friend of Mr. McCain. “The president doesn’t forget easily these slights [from Mr. McCain]. They had actually a lot more in common than they both wanted to admit. John was a great American hero, and nobody can take that away from him.”

The president made an about-face after two of the nation’s largest veterans groups called on Mr. Trump to show more respect to Mr. McCain by ordering all U.S. flags to be flown at half-staff, including at the White House.

The American Legion and AMVETS urged the White House to follow long-established tradition following the death of prominent government officials and keep flags flying at half-staff until burial.

American Legion National Commander Denise Rohan reminded Mr. Trump in a letter that he issued presidential proclamations this year calling for flags to fly at half-staff for extended periods to mark the deaths of former first lady Barbara Bush and the Rev. Billy Graham.

“Sen. John McCain was an American hero and cherished member of the American Legion,” Ms. Rohan wrote. “As I’m certain you are aware, he served five and a half years as a prisoner of war in North Vietnam and retired from the U.S. Navy at the rank of captain. He then served in the U.S. Congress for more than three decades.

“On the behalf of the American Legion’s two million wartime veterans, I strongly urge you to make an appropriate presidential proclamation noting Sen. McCain’s death and legacy of service to our nation, and that our nation’s flag be half-staffed through his interment,” she wrote.

AMVETS said Mr. McCain deserved a longer-lasting show of respect from the White House.

“It’s outrageous that the White House would mark American hero John McCain’s death with a two-sentence tweet, making no mention of his heroic and inspiring life,” Joe Chenelly, AMVETS national executive director, said in a statement. “By lowering flags for not one second more than the bare minimum required by law, despite a long-standing tradition of lowering flags until the funeral, the White House is openly showcasing its blatant disrespect for Sen. McCain’s many decades of service and sacrifice to our country as well as the service of all his fellow veterans.”

In the proclamation, the president said he was ordering flags lowered until sunset Sunday “as a mark of respect for the memory and longstanding service of Sen. John Sidney McCain III.”

In a posthumous farewell statement to the nation, Mr. McCain delivered some final barbs at the president, urging Americans not to “hide behind walls.”

“We weaken our greatness when we confuse our patriotism with rivalries that have sown resentment and hatred and violence in all the corners of the globe,” Mr. McCain said in the statement read by spokesman Rick Davis. “We weaken it when we hide behind walls rather than tear them down, when we doubt the power of our ideals rather than trust them to be the great force for change they have always been.”

The White House greeted the dawn Monday with its flags raised to full staff, after having lowered them to half-staff in the hours after Mr. McCain’s death Saturday night at age 81 from brain cancer.

The short show of respect was exactly the duration proscribed by U.S. code for a member of Congress. But the veterans groups and others criticized Mr. Trump for failing to quickly issue the proclamation to extend the official mourning period.

Ohio Gov. John R. Kasich, a Republican, called the flags flying at full staff “shameful.”

Flags at the Capitol remained at half-staff, and Senate Majority Leader Mitch McConnell, Kentucky Republican, and Minority Leader Charles E. Schumer, New York Democrat, asked the Defense Department to make sure all government flags were lowered as well.

But Sen. James M. Inhofe, Oklahoma Republican, said the flag controversy was partially Mr. McCain’s fault and the result of Mr. Trump and Mr. McCain both being “strong-willed people.”

“Well, you know, frankly, I think that John McCain is partially to blame for that because he is very outspoken. He disagreed with the president in certain areas and wasn’t too courteous about it,” said Mr. Inhofe, who served on the Senate Armed Services Committee with Mr. McCain and repeatedly called the late senator his hero.

Mr. Trump, who had clashed with Mr. McCain since nearly the start of his campaign in 2015 over the direction and image of the Republican Party, ignored reporters’ questions about the senator Monday. The president reportedly directed his staff earlier to withhold the statement of tribute that the White House had prepared to release upon Mr. McCain’s death.

Before he died, Mr. McCain made it known that he didn’t want Mr. Trump to attend his funeral. Former Presidents George W. Bush and Barack Obama, who both ran against Mr. McCain for president, will deliver eulogies Saturday at his request in a service at the Washington National Cathedral.

Mr. Trump’s relationship with Mr. McCain wasn’t always at rock bottom. In May 2016, Mr. Trump told radio host Don Imus, “You know, frankly, I like John McCain, and John McCain is a hero. Also, heroes are people that are, you know, whether they get caught or don’t get caught, they’re all heroes as far as I’m concerned. And that’s the way it should be.”

They endorsed each other for election that year, even after Mr. McCain blasted Mr. Trump over his criticism of the parents of Army Capt. Humayun Khan, who was killed in action in Iraq. The soldier’s father spoke at the Democratic National Convention that year and accused Mr. Trump of anti-Muslim policy proposals.

In August 2016, Mr. Trump said in his endorsement of the senator, “I hold in the highest esteem Sen. John McCain for his service to our country in uniform and in public office.”

But Mr. McCain withdrew his support of Mr. Trump in October 2016 after the publicizing of the “Access Hollywood” tape in which Mr. Trump was heard talking in vulgar terms about women in a 2005 recording.

Mr. Trump then slammed Mr. McCain on Twitter, tweeting, “The very foul-mouthed Sen. John McCain begged for my support during his primary (I gave, he won), then dropped me over locker room remarks!”

In July 2017, Mr. McCain further angered Mr. Trump by casting the pivotal “no” vote on a Republican-led repeal of the Affordable Care Act, providing the decisive vote that kept Obamacare afloat. For nearly a year afterward, Mr. Trump reminded supporters at his rallies of what he considered an irresponsible vote, usually without mentioning Mr. McCain’s name.

When Mr. Trump signed the National Defense Authorization Act named for Mr. McCain as a tribute this month, the president pointedly refused to speak the senator’s name at the signing ceremony at a military base in upstate New York.

Mr. Ruddy, CEO of Newsmax Media, said that despite their well-publicized differences, Mr. McCain and Mr. Trump were on the same sides of many issues, including tax cuts, deregulation and rebuilding the military.

“Sometimes we all want to have a soap opera,” Mr. Ruddy said. “The president is doing a lot of things that John has championed for a long time. And John voted for a lot of stuff that the president has pushed through. His service to the country was distinguished and very rarely paralleled by other people. The country has suffered a tremendous loss.”

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