"We had a lot of family fights", Biden said, adding that McCain often visited his home in Wilmington, Delaware, and they had a great personal relationship outside of the Capitol. John McCain this week, Arizona Cardinals wide receiver Larry Fitzgerald lost a friend.
The public will file through the Capitol's Rotunda past his coffin for six hours to pay their respects.
McCain's widow Cindy, his seven children and his 106-year-old mother Roberta McCain joined scores of members of Congress, state governors, diplomats and other dignitaries at the somber Rotunda ceremony.
Though expected after a long battle with brain cancer, McCain's death rocked Washington last Saturday.
The benediction was delivered by Rear Admiral Barry Black, the Senate chaplain who referred to McCain as a drum major: "May the life of this American patriot inspire us to see you more clearly, to love you more dearly, and to follow you more almost each day".
The service unfolded at North Baptist Church after a motorcade bearing McCain's body made its way from the state Capitol past Arizonans waving American flags and campaign-style signs for the lawmaker. His family watched from the steps.
Similar themes likely will be addressed when Pence, Ryan and Senate Majority Leader Mitch McConnell deliver remarks in the Rotunda.
Pence says, "In every country there are those who put country first, who prize service ahead of self, who summon idealism from a cynical age".
"This is one of the bravest souls our nation has ever produced", Ryan said.
"With John, it was never feigned disagreement". But his support for limited government, for tax reform and support for our armed forces surely left our nation more prosperous and more secure.
Vice-President Mike Pence told mourners Mr Trump had asked him to attend to pay his respects.
McConnell said McCain "would fight tooth and nail for his vision of the common good". Later, she sat at her husband's desk with Sen.
"This whole idea the world should stop for John McCain's funeral is just a disconnect between the Republican Party and the president's priorities - including keeping the House and avoiding impeachment - and the D.C. echo chamber", said Sam Nunberg, a former Trump aide, was quoted as saying by Politico.
It's the first of two days of services honoring the Arizona senator in the nation's capital where he served for more than 30 years in elected office. They'll continue through his the memorial service in Phoenix, his funeral Saturday at Washington National Cathedral and his interment at the Naval Academy cemetery in Annapolis on Sunday.
Other officials represented the administration in Trump's hard-to-miss absence. Earlier this week, the American flag atop the White House that had been at half-staff was raised back up, then lowered again after Trump drew fire from Congress and veterans.
This abiding schism between two larger-than-life figures, both fighters by nature, no doubt helps explain some of the negativity on social media by Trump supporters after McCain died.
A chaplain began the ceremony with a prayer remembering McCain's military service and willingness to work with those he disagreed with. (Tom Williams/CQ Roll Call) Former Sen. "John McCain was such a man".
President Donald Trump, who had feuded bitterly with the senator, was notably absent. 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".