And he said he had authorized military assistance for the funeral and burial.
Before Mr Trump's statement on Monday afternoon, his only commentary on Mr McCain's death had been a perfunctory tweet on Saturday.
US code calls for flags to be lowered in the event of the death of a member of Congress "on the day of death and the following day". Former President Barack Obama issued such an order for the late Sens.
The episode was the latest involving Trump and the powerful symbolism of the American flag.
"The flag of the United States shall be flown at half-staff on all buildings, grounds, and naval vessels of the Federal Government in the metropolitan area of the District of Columbia on the day of death and on the following day upon the death of a United States Senator, Representative, Territorial Delegate, or the Resident Commissioner from the Commonwealth of Puerto Rico, and it shall also be flown at half-staff on all buildings, grounds, and naval vessels of the Federal Government in the State, Congressional District, Territory, or Commonwealth of such Senator, Representative, Delegate, or Commissioner, respectively, from the day of death until interment".
Ansoff said the White House didn't break any rules when it raised the flag to full staff, but that it was "disappointing" to see partisanship influence his decision.
David Popp, a spokesman for McConnell, confirmed his involvement, saying it was a standard request when a senator dies. After criticism, the flag returned to half-staff Monday afternoon. Conservative commentator Bill Kristol, meanwhile, wrote on Twitter: "The White House is now an island of bitterness and resentment in a nation united in respect and appreciation". McCain was also alarmed by the rise of far-right populism in Europe, and the British break with the European Union that Trump has heralded.
Following McCain's death, admirers have recalled with fondness his insistence on fair and civil discourse - in contrast with the frequent incivility of the United States president and his habit of coining unflattering descriptions of political foes.
President Trump did eventually release a statement regarding McCain's death, stating that he "respected Senator John McCain's service to our country" and would be providing military transportation of his remains from Arizona to Washington D.C.
McCain worked across the political aisle to advocate for campaign finance reform with former democratic USA senator Russell Feingold and immigration reform alongside former democratic US senator Ted Kennedy.
Husband: Woman freed temporarily from Iranian prison
Mrs Zaghari-Ratcliffe was arrested in April 2016 at the airport in Tehran after visiting her family with her young daughter. Although the initial release is for three days her lawyer is hopeful that this can be extended, her husband said .
Later Monday evening at a dinner with evangelical leaders, Trump made his first public comments since the senator's death Saturday.
Donald Trump long refused to join the bipartisan group of politicians praising the life of John McCain - a group that now even includes McCain's former captor in Vietnam's infamous "Hanoi Hilton".
He made a decision to end treatment due to "the progress of disease and the inexorable advance of age", according to a released statement from McCain's family.
The statement said other members of the Trump administration, including his chief of staff and secretary of defence, will attend instead. Shortly before Trump issued his written statement, the flag was lowered again to half-staff.
Ted Lempert, a political science lecturer and former politician, said McCain was a conservative with principles, "willing to put the country ahead of politics".
McCain, 81, died Saturday after a battle with an aggressive form of brain cancer that lasted more than a year.
Trump is not expected to attend the funeral or memorial services in Washington for McCain, a senior White House official said. McCain's plane was shot down over Hanoi in 1967 during a bombing run.
It was no secret that the president and McCain frequently clashed. In the years since, McCain was Trump's loudest Republican critic, especially as the president disrupted America's long-time alliances.
McCain took a final swipe at Trump in his farewell message to the nation, delivered posthumously on Monday, denouncing "tribal rivalries".