McCain spoke at the Kennedy Institute's opening in 2015, and remembered his colleague as a "problem solver".
The death of Senator McCain on Saturday was the last of such wars.
Meanwhile, President Trump issued a two line tweet expressing his sympathies to the McCain family.
The comment was restrained as Trump tweets go, yet fell far short of the statements the White House typically releases when a prominent politician dies, regardless of their party.
McCain was the son and grandson of admirals and followed them to the US Naval Academy. Not one. It's been quite a ride. He fought for his country in war and in peace with a dedication that was awe-inspiring. "I made a small place for myself in the story of America and the history of my times". "Thank you Senator McCain for your service to the nation".
But it was just as notable when he held his sharp tongue, in service of a party or political gain. "We'll set a new standard for transparency and accountability with hard work, strong faith and a little courage", McCain said. There will be a service for McCain in Arizona this week, a Republican close to the McCain family told CNN.
Huckabee told Judge Jeanine Pirro of a time on the 2008 campaign trail when McCain - who was deeply interested in cooking - had to be asked to leave a Little Rock, Arkansas, barbecue restaurant in order to continue campaigning for the presidency.
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The McCain-Trump relationship grew heated in 2015 when McCain said Trump's candidacy had "fired up the crazies".
Flake also said that he "admired" McCain's dramatic "no" vote on repealing the Affordable Care Act in July of past year despite voting the other way, as well as McCain's emphasis on senators working together across party lines.
"John McCain was a man of honor, a true patriot in the best sense of the word".
Our thoughts and prayers go out to Senator McCain's family and friends in these turbulent times.
Arizona operatives speculate that one of two former congressmen from the state, John Shadegg and Matt Salmon, could fill the seat.
On October 1967, McCain was on his 23rd bombing round over North Vietnam when he was shot out of the sky and taken prisoner. While in captivity, he suffered torture that left him with lasting disabilities.
When McCain's Vietnamese captors offered him early release as a propaganda ploy, McCain refused to play along, insisting that those captured first should be the first set free.
"On Oct 26, 1967, at Truc Bach Lake, the military and people of Hanoi arrested Major John Sidney McCain, a pilot of the American Navy's air force", it says on the sculpture, which depicts an airman with his hands above his head in front of a broken plane wing.