Mr. McCain's remains will lie in state at Arizona's capitol on Wednesday, before a public viewing Friday in the rotunda of the US Capitol - an honor reserved for the likes of John F. Kennedy, Ronald Reagan and civil rights champion Rosa Parks.
It came after growing outcry over Mr Trump's lack of praise for the 81-year-old former Republican presidential candidate, with whom he frequently clashed ever since his 2016 presidential bid.
John McCain's death then raised back up on Monday in keeping with the United States flag code but to an outcry from the media and others who felt a flag proclamation from President Trump was in order.
And Trump's re-election campaign also released a statement offering condolences and urging Americans to remember McCain and his family - words that did not explicitly praise the senator.
White House officials - chief of staff John Kelly and press secretary Sarah Sander - had planned to put out a more detailed statement, and one that described McCain as a war-hero.
Schumer also called on Republicans to join him in his efforts to rename a Senate office building, now named for the controversial former Sen. "Cause most of the ones who would do the wrong thing by John McCain didn't have the guts to do the right thing when it was their turn".
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In Vietnam, people went to the US Embassy to pay their respects to McCain who had played an important role in normalizing diplomatic relations with the US in 1995.
Trump's White House lowered its flag on Saturday, then raised it back following the minimum period under law.
Trump and McCain were frequent critics of each other.
As he prepared for the end of his life, McCain said he did not want Trump at his funeral, instead favoring attendance by Pence, with whom he served in Congress. We never hide from history. He will deliver a reading at the Phoenix North Baptist Church ceremony for the passing of Fitz's friend, Senator John McCain.
"We weaken our greatness when we confuse our patriotism with tribal rivalries that have sown resentment and hatred and violence in all the corners of the globe", McCain wrote in the letter.
At a dinner Monday night, Trump included a brief mention of McCain in his remarks, saying "we very much appreciate everything that Senator McCain has done for our country".
Arizona Gov. Doug Ducey (R) has said he will wait until after McCain's funeral Sunday to appoint a successor to serve out the remainder of the senator's term through 2020. "I would have been sixth from the bottom, not fifth", Graham said he would say.