The last visit to Washington by McCain - who died last Saturday at age 81 after a yearlong battle with cancer - is being spread out over two days with former presidents George W. Bush and Barack Obama to eulogize him during a memorial service Saturday at the city's National Cathedral.
Biden, a Democrat who was among the friends the Republican senator made across the aisle, said he thought of McCain as a brother, "with a lot of family fights".
"Many people might wonder what a young African-American kid from Minnesota and a highly decorated Vietnam hero turned United States senator might have in common", Fitzgerald said.
On Thursday, former Vice President Joe Biden remembered McCain as a brother, and said the two were "cockeyed optimists" in a memorial service for McCain at a church in Phoenix.
An honor guard of soldiers, sailors, airmen and Marines carried McCain's flag-draped casket up the Capitol's East steps about 10:45 a.m., as heavy rain fell.
Mr McCain's longtime chief of staff, Grant Woods spoke about the senator's "terribly bad driving" and his sense of humour - which drew a chuckle.
Fitzgerald recalled how McCain once took him on a tour of the U.S. Senate. He talked about McCain's character and how he parted company with those who "lacked the basic values of decency and respect, knowing this project is bigger than yourself".
Biden said McCain "could not stand the abuse of power wherever he saw it, in whatever form, in whatever country".
People carrying campaign style placards with the word "McCain" and holding US flags lined part of the part of the 8-mile (13-kilometer) route the motorcade was taking Thursday to the North Phoenix Baptist Church.
More than 30 current and former senators (Republicans and Democrats, alike), Arizona Gov. Doug Ducey, local officials, Arizona sports stars and around 1,000 Arizonians paid their respects at the service at the North Phoenix Baptist Church. At one point, Cindy McCain, the senator's wife of 38 years, rolled down a window and waved back.
At the end of the almost 90-minute ceremony, McCain's casket was wheeled out of the church to "My Way", in tribute to a politician known for following his own path.
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President Donald Trump ― who was not invited to McCain's memorial ― said in a statement Monday that he asked Vice President Mike Pence to give an address at the ceremony.
John McCain today in Phoenix, Arizona, and described the meaningful friendship that was formed between the two men who came form very different worlds. John McCain in Phoenix, Ariz.
"I think he's a good measure of all of us". "I do everything I can to avoid contact", Fitzgerald joked. "I have a very strong opinion, all right".
"Generations of Americans will continue to marvel at the man who lies before us, the cocky, handsome naval aviator who barely scraped through school, and then fought for freedom in the skies; who witnessed to our highest values, even through awful torture; and who became a generational leader in the United States Senate, where our nation airs its great debates", McConnell said.
The memorial was laced with humor and featured a racially and ethnically diverse roster of speakers and other participants.
Biden's remarks were similar to a spirited McCain speech on the Senate floor last summer, in which the Arizona Republican begged his colleagues to restore to the chamber a spirit of compromise that had helped forge national greatness.
Fitzgerald, who was a personal friend of McCain's, said he was the "authentic American hero".
As the hearse made its way along the 8-mile route, people held signs that read simply "McCain" and cars on the other side of the highway stopped or slowed to a crawl in apparent tribute.
A few firefighters saluted from atop a fire engine parked on an overpass as the motorcade passed underneath on Interstate 17.
"Let us go now from this place in peace and comforting one another in the sure knowledge that one day we will join John in his home in Heaven", Father Reese said.