McCain's words were a clear reference to Trump's infamous 2016 campaign slogan, "Make America Great Again", and the subsequent rallying cry for his base.
Trump, who has feuded publicly with McCain and mocked his military service, received five deferments that got him out of serving in the military.
The six-term senator and 2008 Republican presidential nominee John McCain.
12 event, new iPhones expected
A woman looks at the screen of her mobile phone in front of an Apple logo outside its store in Shanghai, China July 30, 2017. It's now unclear if Apple will save this announcement for later in the year or reveal all alongside its new range of phones.
The casket of the late US Senator John McCain, Republican of Arizona, is carried out after the National Memorial Service at the Washington National Cathedral in Washington, DC. John McCain, R-Ariz., at Washington National Cathedral in Washington, Saturday, Sept. 1, 2018.
Bush said whatever rivalry he had with Sen. Those critics, she said, still have an opportunity to emulate her father's legacy. "Like the hero of his favorite book, John McCain took the opposite view", she said.
In a video clip shared on Twitter, Bush takes the small parcel from Laura Bush, and passes it to Obama.
The speakers seemed to criticize the president's policies and politics, if not Trump himself. "It's a politics that pretends to be courageous, but in fact is born of fear". "To the face of those in authority, John would insist, we are better than this".
"We are better than this, America is better than this", Bush lamented.
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Friday morning, Franklin's open casket was placed at the front of the church as mourners streamed by in advance of the service. Former president Bill Clinton , Rev Jesse Jackson and Smokey Robinson will all speak at the Queen of Soul's final send-off.
An honor guard team carries the flag-draped casket bearing John McCain from at the National Cathedral, Saturday.
Mr Bush said one of the great gifts in his life was becoming friends with his former White House rival.
They included Hollywood actor and liberal political activist Warren Beatty; former independent NY mayor Michael Bloomberg; liberal former senator Russ Feingold, who drew up campaign finance reform legislation with McCain; and Russian opposition politician Vladimir Kara Murza. Russ Feingold of Wisconsin and former chief of staff to McCain Mark Salter will also serve as pallbearers, among others. Obama said, referencing McCain's request that George W. Bush also speak at the event. Thousands of Americans lined up outside the U.S. Capitol in stifling heat to say a final goodbye as he lay in state.
An outpouring of love and memories flooded in immediately following his passing and the decorated Vietnam War veteran was later honored in Arizona on Wednesday, the day he would've turned 82. McCain is to be buried Sunday in a private ceremony at the U.S. Naval Academy Cemetery.
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The president has not named Jones - whose reputation he once described as "amazing" - in any of his tweets on the issue. That being said, if you're going to make an allegation that there's a big conspiracy, you should do your due diligence.
Bush praised McCain for standing up for the working-class Americans over the interests of Washington, D.C.