Later on in the day, McCain's casket will be flown to Washington D.C. from Phoenix. The Arizona Republican, whose relationship with President Trump has always been fraught, died Saturday at the age of 81 after battling brain cancer. McCain will lie in state in the Arizona State Capitol and then at the U.S. Capitol Rotunda.
Flags at the White House are lowered back to half-staff Monday afternoon.
The White House, unusually, has issued no statement on the death, with Trump tweeting only a brief statement of condolence for the family, with no mention of McCain's military and political careers.
Finally, under pressure from veterans and members of Congress, Trump said in a statement later on Monday that he respected McCain's service to the nation and had ordered flags to half-staff.
"Sen. John McCain made an incredible, positive impact on our country and on all who call it home".
A private funeral is planned for Sunday afternoon at the Naval Academy Chapel followed by a private burial at the academy cemetery.
During several events open to cameras Monday, Mr. Trump did not answer shouted questions from reporters about his response to McCain's death.
After Democratic Ted Kennedy of MA died in 2009, President Barack Obama ordered flags at the White House flown at half-staff for five days.
McCain was a frequent Trump critic and his family has said he did not want the president to attend his funeral.
The statement, drafted before the senator's death Saturday, would have commended the Arizona Republican for his military service and his decades in the Senate. McConnell says serving with McCain "was never a tiresome affair".
I have asked Vice President Mike Pence to offer an address at the ceremony honoring Senator McCain at the United States Capitol this Friday.
Obama, who triumphed over McCain in 2008, said that despite their differences, McCain and he shared a "fidelity to something higher - the ideals for which generations of Americans and immigrants alike have fought, marched, and sacrificed".
Other tributes poured in from around the globe. "He devoted his entire life to his country".
"Senator McCain will long be remembered for his years of dedicated service to our nation", said Gov. Colyer, "He displayed great courage and strength in the way he lived his life. It sprang from his belief in democracy and freedom".
Chancellor Angela Merkel called McCain: "A tireless fighter for a strong trans-Atlantic alliance". In October 1967, the then US Navy pilot McCain parachuted out of his Skyhawk jet to be taken prisoner and spend more than five years as a prisoner of war.