McCain died on Saturday, four days shy of his 82nd birthday, following a year-long battle with an aggressive form of brain cancer.
Richard Cordray, the Democratic nominee for governor and a former OH attorney general, said on Twitter, "Saddened to hear of the passing of John McCain - a hero who sacrificed more for our country than many of us ever could imagine".
"Senator John McCain was an American patriot and hero whose sacrifices for his country, and lifetime of public service, were an inspiration to millions".
McCain quickly snatched the microphone away and emphatically said, "No, ma'am. The League sends our condolences to the entire McCain family-a family itself the picture of diversity and inclusion-during this hard time".
John McCain, the war hero who became the GOP's standard-bearer in the 2008 election, has died. A sign near the driveway of his home read, "Sen McCain, thank you for your service". Before his death, McCain's wife Cindy said in a tweet that the McCain family was "overwhelmed by the outpouring of love and support from around the world".
Former presidents, including those who blocked Mr McCain's own White House ambitions, offered emotional tributes.
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"But we shared, for all our differences, a fidelity to something higher - the ideals for which generations of Americans and immigrants alike have fought, marched and sacrificed".
Mr Obama said they "saw our political battles, even, as a privilege, something noble, an opportunity to serve as stewards of those high ideals at home, and to advance them around the world".
French President Emmanuel Macron tweeted in English that McCain "was a true American hero".
Former president George W Bush, who defeated Mr McCain for the 2000 Republican presidential nomination, called Mr McCain a "man of deep conviction and a patriot of the highest order" and a "friend whom I'll deeply miss". He devoted his entire life to his country. "It sprang from his belief in democracy and freedom".
"Senator McCain was one of the great political figures of our time and a tireless fighter for a strong transatlantic alliance; his importance stretches far beyond his own country".
The son and grandson of four-star admirals, McCain will be buried in the cemetery of the US Naval Academy in Annapolis, on a peaceful hillside overlooking the Severn River.
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Mr Trump has refused to publish a letter drafted by White House staff on the death of Mr McCain. He had an entire career, an entire life, before Trump took office previous year .
"Senator McCain also told us about the time he spent in Subic as a young Navy aviator serving in Vietnam and his arrival at Clark Air Base after his release as a prisoner of war".
John McCain was close friends with Senator Lindsey Graham and former senator Joe Lieberman.
He was elected to the House of Representatives in 1982 and the Senate in 1986.
Senate Minority Leader Chuck Schumer, who lauded his colleague as a "truth teller", says he wants to rename one of the Senate office buildings after McCain "so that generations remember him".
"As you go through life, you meet few truly great people".
In Phoenix, a succession of ordinary Arizonans left flags or flowers at McCain's office or a local mortuary to honour the man who represented them in Congress for 35 years.
Michigan Gov. Rick Snyder lowers flags to honor Sen. John McCain
Bush and Barack Obama , who blocked McCain's own White House ambitions, are among those expected to speak at McCain's funeral. McCain will lie in state at the Arizona Capitol before being honored with a memorial service on Thursday.