Mr McCain died on Saturday surrounded by his family, a short statement released by his office said.
He was diagnosed with an aggressive brain tumour in July 2017 and had been undergoing medical treatment.
His family had announced that Mr McCain, who left Washington in December, had decided to stop treatment on Friday.
Mr McCain’s daughter Meghan said the task of her lifetime would now be “to live up to his example, his expectations, and his love.
“The days and years to come will not be the same without my dad – but they will be good days, filled with life and love, because of the example he lived for us,” she wrote in a statement shared on Twitter.
The six-term senator and 2008 Republican presidential nominee was diagnosed after doctors discovered his tumour during surgery to remove a blood clot from above his left eye last July.
His family said he would lie in state in Phoenix, Arizona, and in Washington DC before a funeral at the Washington National Cathedral and his burial in Annapolis, Maryland.
The son and grandson of Navy admirals, Mr McCain was a fighter pilot during the war in Vietnam. When his plane was shot down, he spent more than five years as a prisoner of war.
While in the custody of his captors, he suffered torture that left him with lasting disabilities.
McCain fought hard to the very end
By Anthony Zurcher, BBC North America reporter
John McCain was born on the eve of World War II, at the dawn of the “American Century” – a time when the US was at the peak of its political, military and cultural power. He dies at what could be considered that age’s twilight, as the nation turns inward and contemplates walls, literal and metaphorical, to insulate itself from the rest of this world.
The life of the senator from Arizona marked the arc of this journey.
He suffered, as the nation suffered, from the morass of Vietnam.
As a young politician he was tempted by the lure of power and money, caught up in an influence-peddling scandal that nearly cost him his career.
In his first run for president in 2000, he capitalised on an anti-establishment sentiment and longing for authenticity that would later come to crest with Donald Trump’s election. In 2008, he won the Republican nomination, only to see his hopes dashed by the phenomenon that was Barack Obama and a crumbling US economy.
McCain never won the top political office for which he longed. Throughout his life, however, he offered a full-throated defence of an America that was active and engaged in the world. In his final years he sparred with Mr Trump over the direction of the Republican Party and the principles it should embrace.
It’s an open question as to whether these views have a future in his party. McCain, however, fought for what he believed was right until the very end. Agree with him or not, that is undeniably a most appropriate epitaph.
Tributes began to pour in for Mr McCain as soon as the news of his death was announced.
Donald Trump, whom Mr McCain has strongly criticised, tweeted: “My deepest sympathies and respect go out to the family of Senator John McCain. Our hearts and prayers are with you!”
Sarah Palin, who was Mr McCain’s running mate during his 2008 bid for president, said the world had lost “an American original”, sharing a picture of herself with the man she called her friend.
John McCain was my friend. I will remember the good times. My family and I send prayers for Cindy and the McCain family.
– Sarah Palin and family pic.twitter.com/KRvcIQ99cA
— Sarah Palin (@SarahPalinUSA) August 26, 2018
Tributes were also paid from the other side of the political spectrum.
Barack Obama, the Democrat who beat Mr McCain to the presidency, said despite their differences, they shared “a fidelity to something higher – the ideals for which generations of Americans and immigrants alike have fought, marched and sacrificed.
“Few of us have been tested the way John once was, or required to show the kind of courage that he did,” Mr Obama said. “But all of us can aspire to put the courage to put the greater good above our own.
“At John’s best, he showed us what that means.”
Former vice-president, long-time friend and political opponent Joe Biden said Mr McCain’s “impact on America hasn’t ended”.
“John McCain’s life is proof that some truths are timeless,” he said in a statement. “Character. Courage. Integrity.
“A life lived embodying those truths casts a long, long shadow. John McCain will cast a long shadow.”