The service at Washington National Cathedral is one of several events honoring McCain's life this week.
Bush on Saturday hailed John McCain as "unequaled" in a eulogy for the late senator.
After his body arrived at the cathedral, McCain's daughter Meghan McCain opened the memorial service with a tearful and impassioned tribute to him.
McCain asked former Presidents Barack Obama and George W. Bush to speak at the service to highlight the bridge-building that McCain espoused.
Two former presidents, a pre-eminent diplomat and a former Senate colleague are among the speakers on the schedule for Saturday's memorial service for John McCain at Washington National Cathedral.
"John was, above all, a man with a code", Bush said.
Bush said that McCain always recognized that his opponents "were still patriots and human beings".
He added that, "To the face of those in authority, John McCain would insist that we are better than this, America is better than this".
John McCain is set to receive a final presidential farewell, but just not from the current president. "In the end, I got to enjoy one of life's great gifts, the friendship of John McCain".
Among those who served as pallbearers included actor Warren Beatty, Russian dissident Vladimir Kara-Murza, former Vice President Joe Biden, former New York City Mayor Michael Bloomberg, FedEx founder Fred Smith and billionaire investor Ronald Perelman. Russ Feingold of Wisconsin and former chief of staff to McCain Mark Salter are also serving as pallbearers, among others.
Members of the military slowly carried McCain's coffin into the cathedral, past friends and loved ones who have counted among the top power players in US politics for decades.
McCain died of brain cancer on August 25 at age 81.
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 will be laid to rest next to his Naval Academy classmate and lifelong friend, Admiral Chuck Larson, who passed away in 2014.
McCain denounced him for using language that "fired up the crazies", while Trump said McCain was a "dummy" who had barely managed to graduate from the US Naval Academy.
Saturday's service is the last for McCain in Washington. Many interpreted the bipartisan funeral as a political message aimed at President Donald Trump, who was not invited to the ceremony and who had an adversarial relationship with the late senator.
He went on to attack McCain's service in the military, saying of the onetime prisoner of war: "I like people that weren't captured".