WASHINGTON — In the Senate chamber on Monday, John McCain’s desk was draped in black and topped with a vase of white roses. The majority leader, Senator Mitch McConnell, rose to praise Mr. McCain as a colleague and hero who “spotlighted many of our highest values.” Outside, an impromptu memorial took shape as the flags over Capitol Hill flew at half-staff.

In only one building in Washington were Mr. McCain’s legacy and achievements greeted with anything like ambivalence: the White House.

President Trump, under enormous public and private pressure, finally issued a proclamation of praise for Mr. McCain on Monday afternoon, two days after the senator’s death, and ordered the flag to be flown at half-staff seemingly in the only place it wasn’t already, the presidential complex.

The day had begun with the remarkable sight of the flag flying atop the White House’s flagpole, while just beyond the building, at the Washington Monument, others fluttered midway down the poles that circle the obelisk. The president stubbornly refused repeated requests from officials as senior as Vice President Mike Pence and John F. Kelly, the White House chief of staff, to acknowledge Mr. McCain’s death with a formal and unifying statement, according to four administration officials who spoke on condition of anonymity to describe private conversations.

At midday, the drama was punctuated by the words of Mr. McCain himself, whose final statement to the nation was delivered posthumously through a top aide.

“We weaken our greatness when we confuse our patriotism with tribal rivalries that have sown resentment and hatred and violence in all the corners of the globe,” Mr. McCain wrote in a statement delivered by Rick Davis, his family spokesman and former campaign manager. “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.”

[Read Senator John McCain’s farewell statement.]

Then, after the ire from critics, lawmakers and veterans’ groups crescendoed, the president released a statement to try to put the matter behind him, but it began with highlights of past conflicts.

“Despite our differences on policy and politics, I respect Senator John McCain’s service to our country,” Mr. Trump said, “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.”

On Monday evening, he told a dinner of evangelical supporters, “We very much appreciate everything that Senator McCain has done for our country.”

The president, who Mr. McCain had previously said was not invited to his funeral services, said other senior aides would attend memorial events in his place. They will be Mr. Kelly; John R. Bolton, the national security adviser; and Defense Secretary Jim Mattis. Along with Mr. Kelly and Mr. Bolton, Sarah Huckabee Sanders, the White House press secretary, and Bill Shine, the deputy chief of staff for communications, had urged the president to let aides take over the White House response to Mr. McCain’s death.

For much of the day, Mr. Trump appeared oblivious to the criticism. He resisted when Mr. Kelly called him at 7 a.m. and lobbied him to let the staff handle the response, a person familiar with the exchange said. He let Mr. Kelly know the day and week would continue as scheduled. Then he tweeted about professional sports, trumpeted a revamped trade deal with Mexico and hosted the Kenyan president.

At one point, Mr. Trump seemed so eager to publicly deliver a policy victory on the trade deal that television cameras inside the Oval Office went live before the telephone equipment had Enrique Peña Nieto, the Mexican president, on the line.

“Enrique?” Mr. Trump asked, growing flustered on live television as his aides tried to figure out the phone. “Do you want to put that on this phone, please? Hello? Be helpful.”

With the president’s attention elsewhere, the visual of the flag raised high at the White House made the rounds on social media, and prompted a statement from the American Legion, the nation’s largest wartime-veterans service organization.

“On the behalf of the American Legion’s two million wartime veterans, I strongly urge you to make an appropriate presidential proclamation noting Senator McCain’s death and legacy of service to our nation, and that our nation’s flag be half-staffed through his interment,” Denise Rohan, the national commander of the American Legion, wrote in a statement that one White House adviser said caught the president’s attention.

As he publicly dodged questions from reporters about Mr. McCain, Mr. Trump allowed a split screen to unfold as much of the nation, including Mr. McCain’s Senate colleagues, publicly memorialized him on Capitol Hill. Senators rose one by one to pay their respects to a man they called a statesman and hero.

“Generation after generation of Americans will hear about the cocky pilot who barely scraped through Annapolis, but then defended our nation in the skies,” Mr. McConnell said in uncharacteristically personal remarks. “Witnessed to our highest values even through terrible torture. Captured the country’s imagination through national campaigns that spotlighted many of our highest values. And became so integral to the United States Senate, where our nation airs and advances its great debates.”

Senator Jeff Flake, Mr. McCain’s fellow Arizona Republican, implored his colleagues to learn something from Mr. McCain’s iconoclasm.

“We have lately wasted a lot of words in this town doing and being everything that John McCain was not,” he said. “We would do well to allow this moment to affect us in ways reflected not merely in our words but also our deeds.”

Senators from both parties appeared to embrace a proposal, first made by Senator Chuck Schumer of New York, the Democratic leader, to rename one of the Senate’s office buildings after Mr. McCain. Doing so could provide senators any easy step around a potentially thorny fight: The building’s current eponym, Richard B. Russell, was a staunch segregationist who led the fight in the Senate against desegregation and the Civil Rights Act of 1964.

Many Democrats but especially Republicans, long since weary of Mr. Trump’s impolitic handling of the duties of his office, offered only passing criticism of the president’s ambivalence. Senator James M. Inhofe, the Oklahoma Republican who occasionally clashed with Mr. McCain and will succeed him as the chairman of the Armed Services Committee, told reporters that the senator was “partially to blame” for Monday’s flag controversy. Mr. McCain, he said, “wasn’t too courteous” in his disagreements with Mr. Trump.

Senator Orrin G. Hatch of Utah, the chamber’s longest-serving Republican, sounded more regretful about the president’s behavior.

“That should not have happened. That should have been automatic,” he said of the president’s delay in issuing a statement. “You just do things that are sensible and sensitive.”

Flag policy has been a priority for Mr. Trump, even before he was president. Mr. Trump has attacked players in the National Football League who kneel during the playing of the national anthem in a silent protest for civil rights, saying their actions were disrespectful to the flag and the United States military.

The president previously issued a proclamation for flags to be lowered on the day that the Rev. Billy Graham was buried early this year. Mr. Trump made the proclamation on Feb. 21, and Mr. Graham was buried in early March.

This time, White House officials were reluctant to wade into the flag symbolism that had been so important to Mr. Trump.

“I’m not commenting on that,” Hogan Gidley, a White House spokesman, said in a brief phone call, adding that he was not working in “flag world.”

Mr. McCain, who had survived a Vietnam War prison camp and weathered the gradual coarsening of politics within his own party, appeared to have a sense of how the day might unfold. For one last time, Mr. McCain stepped in to call for patriotism over politics when Mr. Trump would not.

In his final statement, Mr. McCain alluded to “blood and soil,” a German nationalist slogan dating to the 19th century that has been resurrected by the white nationalist movement in the United States.

“We are citizens of the world’s greatest republic, a nation of ideals, not blood and soil,” Mr. McCain wrote.

He closed his letter with a plea: “Do not despair of our present difficulties but believe always in the promise and greatness of America, because nothing is inevitable here. Americans never quit. We never surrender. We never hide from history. We make history.”