"Despite our differences on policy and politics", it read in part, "I respect Senator 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".
In Vietnam, people went to the US Embassy to pay their respects to McCain who had played an important role in normalizing diplomatic relations with the US in 1995.
McCain's office said a livestream will be available for the services at the National Phoenix Baptist Church and the National Cathedral.
Flags at the U.S. Capitol, where he served two terms in the House and six terms in the Senate, remained at half staff on Monday morning. Senator John McCain was an American hero and cherished member of The American Legion.
"The fact that people say good things about [McCain] now, I appreciate, but the family appreciates it", Graham said, detailing that McCain is survived by his wife, Cindy, seven children and give grandchildren.
Former Vice President Joe Biden, who lost his son Beau to glioblastoma and was helping the McCain family through the senator's illness, said McCain's "impact on America hasn't ended - not even close".
It will conclude almost a week of events honouring the man, whose body will lie in state on Wednesday in the Arizona State Capitol and on Friday in the Capitol Rotunda, Washington DC, with a formal ceremony and time for the public to pay respects.
The American flag files at half-staff at the White House, Monday, Aug. 27, 2018, in Washington.
A family spokesman issued a farewell statement from McCain in which he said of the United States: "We weaken our greatness when we confuse our patriotism with tribal rivalries".
Among the pallbearers will be Democratic former vice president Joe Biden, ex-senator Russ Feingold, former NY mayor Michael Bloomberg and actor Warren Beatty.
Senate Majority Leader Mitch McConnell, a longtime ally of McCain, said that the decision came "in coordination with Speaker of the House Paul Ryan, Senate Democratic Leader Chuck Schumer and House Democratic Leader Nancy Pelosi". "He was one of the most courageous men of the century".
German Chancellor Angela Merkel called McCain "a tireless fighter for a strong trans-Atlantic alliance; his significance went well beyond his own country". But his condition did not stop him from taking the political centre stage, especially by railing against what he saw as the excesses of President Donald Trump. Americans never quit. We never surrender.
Indian Prime Minister Narendra Modi also offered condolences. "I consider my friendships to be friendships". First lady Melania Trump tweeted thanks to McCain for his service to the country.