The proclamation affects the flag atop the White House and all public buildings, as well as military installations and embassies.
Trump is not expected to attend this weekend's funeral of the late US senator McCain.
"The president will not be, as far as we know, attending the funeral. That's just a fact," longtime aide Rick Davis told a press briefing in Arizona.
McCain's posthumous message
Late US senator John McCain took a final swipe at Donald Trump in his farewell message to the nation delivered posthumously Monday, denouncing "tribal rivalries," as an aide confirmed the president will not attend the lawmaker's funeral.
The message came as Trump found himself mired in controversy over his rather conspicuous failure to pay tribute to McCain, who died Saturday at 81 after a year-long battle with brain cancer.
The flag at the White House was at full-staff Monday, and no formal statement came from Trump beyond a brief tweet.
On Monday, McCain offered his final thoughts, in a message read by his former campaign manager Rick Davis -- and he did not spare the Republican president.
"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," McCain said in the statement.
"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," he added -- an apparent reference to Trump's plans for a border wall.
"I lived and died a proud American. We are citizens of the world's greatest republic, a nation of ideals, not blood and soil," the former prisoner of war, two-time Republican presidential candidate and titan of US politics said.
"Do not despair of our present difficulties but believe always in the promise and greatness of America, because nothing is inevitable here," he said, adding that the country "will get through these challenging times."
That appeared to be a reference to Trump's "Make America Great Again" slogan.
Trump won't attend funeral
McCain's death has left many American mourning the loss of a national hero who repeatedly challenged the status quo and consistently sought bipartisan solutions to the country's problems.
The longtime senator clashed repeatedly with Trump despite being from the same party, and the president -- who in 2016 famously dismissed McCain as "not a war hero" -- has paid scant tribute to the senator in the wake of his death.
With bad blood between them at the boil, McCain reportedly excluded Trump from his funeral ceremonies -- a development that Davis confirmed on Monday.
"The president will not be, as far as we know, attending the funeral. That's just a fact," Davis told a press briefing in Arizona.
The two men who defeated McCain in his White House campaigns, Republican George W. Bush and Democrat Barack Obama, are expected to deliver eulogies at a Saturday service at the National Cathedral in Washington, a day after McCain's body lies in state in the US Capitol.
McCain will be buried Sunday at the US Naval Academy in nearby Annapolis, Maryland in a private funeral service.