In the statement, released two days after the statesman's death, the United States president said he respects Mr McCain's "service to our country".
"Fortunately, I've had some time to get prepared and put some of my thoughts on paper", Fitzgerald said.
'Clearly they had a contentious relationship. John McCain on Monday.
"He's not a war hero", Trump said at a Republican presidential forum in 2015. He still refuses to call McCain a war hero. In a display of unity at a time of deepening political, racial and religious divisions, he instead requested former presidents George W Bush and Barak Obama - the Republican and Democrat who beat him in his two runs for the White House - to deliver his eulogy.
But they were flying at full-staff by Monday morning, even as the flags at the US Capitol remained at half-staff.
Trump eventually spoke about the senator while meeting with evangelical leaders at the White House on Monday night.
Senator McCain also expressed his deep gratitude and love of country.
Serving as pallbearers will be former Vice President Joe Biden, former Defense Secretary William Cohen, former New York Mayor Michael Bloomberg, and former Senators Gary Hart, Russ Feingold and Phil Gramm. Lindsey Graham shared some of the late senator's final words to him and elaborated on McCain's legacy.
"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.
Trump campaigned on a promise to build a wall across the US border with Mexico and force Mexico to pay for it.
McCain, 81, died over the weekend after a lengthy battle with cancer.
The White House lowered its US flag to half-staff, raised it back up and on Monday lowered it again after the death of Senator John McCain, in an unusual and confusing break with protocol on the passing of a national leader.
Vice President Mike Pence will speak at a ceremony for McCain on Friday at the U.S. Capitol.
McCain was a frequent Trump critic and his family has said he did not want the president to attend his funeral.
He declined to answer any question's about the flag or Trump's response to McCain's death. John McCain's memorial will be one of the greatest honors he's ever received. "That's just a fact", Davis said, declining to reiterate what has already been reported about the McCain family's wishes.
Ward replied: "I think they wanted to have a particular narrative that is negative to me".