The message was posted about an hour after the memorial service at the Washington National Cathedral ended.
"It's a politics that pretends to be fearless and tough but in fact is born of fear".
It was a set of values that should serve as a reminder, he said: "If we're ever tempted to forget who we are, to grow tired of our cause, John's voice will always come as a whisper over our shoulder: 'We're better than this".
Former Sen. Joe Lieberman of CT, a Democrat who was once re-elected as an independent, also spoke of McCain's patriotism and ability to bring people of all political parties together. While offering "sympathy" and "respect" to the 2008 Presidential candidate's family, the current President didn't make a single remark that day about McCain's long military and political service.
Meghan McCain is saying goodbye to her father.
The casket arrives for the memorial service of US Senator John McCain at National Cathedral in Washington, US, September 1, 2018. "To the face of those in authority, John McCain would insist "We are better than this, America is better than this", Bush said in his tribute. "My father is gone", she said.
Though they come from different political parties - Michelle Obama is married to Barack Obama, a Democrat leader and two-time US President, while George W Bush has been a Republican President of the US - the two are known to share a "real fondness" for each other.
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On Thursday, a memorial service was held in Phoenix and on Friday, a ceremony was held in the US Capitol, where McCain's body was lain in state.
McCain died due to glioblastoma (an aggressive brain form of brain cancer) on August 25 at age 81.
One of the most tense episodes came in July of a year ago, when McCain - shortly after making public his brain cancer diagnosis - joined just two other GOP colleagues in voting against a Republican-proposed bill that would have repealed certain aspects of the 2010 Affordable Care Act, a USA health-care overhaul popularly known as "Obamacare".
Very few non-presidents in American history have had the kind of week remembrance that John McCain had, Brownstein added.
The almost three-hour service was a remarkable show of defiance against a president McCain openly defied in life.
Republican Sen. Lindsey Graham of SC, a close friend of McCain's, and McCain's children Jimmy and Sidney McCain are taking part in the ceremony as well. Ivanka Trump and Jared Kushner also attended. During the 2016 presidential campaign, Mr Trump - who received draft deferments during the Vietnam War - disparaged Mr McCain's prisoner of war status. He refused to be released without his men. The dignitaries included three former presidents, at least four former secretaries of state and members of the House and Senate, where McCain served for more than three decades. McCain pushed for a compromise that included securing U.S. borders and providing a pathway to citizenship for undocumented immigrants. "We never doubted the other man's sincerity or the man's patriotism".
Bush also mourned McCain's passing.