Trump made no comment when McCain chose to end treatment for brain cancer on Friday, and also failed to mention McCain earlier this month when he signed a defence-spending bill named for him. With the Senator when he passed were his wife Cindy and their family.
"Senator McCain exemplified what it means to be a warrior and dedicated public servant", Marine Corps Gen. Joe Dunford, the chairman of the Joint Chiefs of Staff, said in a written release. "This is a rare and distinct occurrence for a truly special man. John McCain is Arizona, and we will honor his life every way we can".
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African-American leaders sent condolences to the family of Sen.John McCain, who leaves a mixed legacy with the Black community.
Still, Obama said he and the senator "shared, for all our differences, a fidelity to something higher - the ideals for which generations of Americans and immigrants alike have fought, marched and sacrificed". "John, you will be missed", the current president of Estonia, Kersti Kaljulaid, tweeted.
"When I heard about the passing of Senator McCain, I felt that It was my duty and responsibility to come out and to stand in his honor", he said. His contributions into the security of the Baltic states will never be forgotten.
Trump said he asked Vice President Mike Pence to offer an address at McCain's ceremony at the U.S. Capitol on Friday. "And the best way to prevent Russian misbehaviour by having a credible, strong military and a strong North Atlantic Treaty Organisation alliance", the senator said.
Republican Arizona Governor Doug Ducey will appoint a member of his own party to succeed McCain after the funeral.
Malloy called McCain a hero who put his life on the line to defend the nation during a time of war, experiencing torture and suffering.
"An American president does not lead the free world by congratulating dictators on winning sham elections", McCain said.
After Trump became president, McCain blasted what he called the president's attempts to undermine the free press and rule of law and lamented the "half-baked, spurious nationalism" of the Trump era.
McCain had long feuded with President Donald Trump, and two White House officials said McCain's family had asked, before the senator's death, that Trump not attend the funeral services.
A private burial service next Sunday will conclude almost a week of events honoring the Navy aviator, prisoner of war, congressman, longtime senator and presidential contender. He was treated periodically with radiation and chemotherapy until he decided not to receive any more treatments on 24 August 2018.
Bush and Barack Obama, who blocked the Arizona Republican's own White House ambitions, are among those expected to speak at McCain's funeral.
Sources said that McCain had requested that George W Bush and Barack Obama deliver the eulogies at his funeral.
The late Senator will lie in state in the Arizona Capitol on 29 August, the day he would have turned 82 years old.