McCain, a prisoner of war in Vietnam, longtime USA senator from Arizona and 2008 Republican presidential nominee, died of brain cancer on Saturday at age 81.
The 81-year-old Senator John McCain who died on Saturday after a long illness left a statement to be read by Rick Davis, his former presidential campaign manager.
On Sunday, he emphasized that the building would be better named to honor McCain, as the late Arizona senator was an "antidote when he saw bigotry".
The American flag at the White House flies at full staff on August 27 in Washington, DC.
During several events open to cameras Monday, Mr. Trump did not answer shouted questions from reporters about his response to McCain's death.
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McCain was a frequent Trump critic and his family has said he did not want the president to attend his funeral. Some believe that as a tribute, flags should remain at half-staff until the deceased is interred.
Over the weekend, Maryland Gov. Larry Hogan, R, ordered the Maryland state flag to be lowered to half-staff in honor of McCain until his burial on Sunday. That prompted many Americans to lower flags to half-staff, a traditional gesture of honor.
After Democratic Ted Kennedy of MA died in 2009, U.S. President Barack Obama ordered flags at the White House flown at half-staff for five days.
The break with precedent of previous presidents - who have typically released effusive official statements for noteworthy Americans upon their death - underscored the bitter relationship between the two men, Trump's continued anger toward McCain, and the substantive and stylistic differences between them, people close to both men said.
"If the president put out a flowery statement about John McCain's life, the media would criticize it and say it's not consistent with the other things he's said in the past, and it would become a story about the president", Short said during an appearance on CNN. McCain, who served in the U.S. Navy, graduated from the academy in 1958. He then served in the U.S. Congress for more than three decades. First lady Melania Trump tweeted thanks to McCain for his service to the country. I also direct that the flag shall be flown at half-staff for the same period at all United States embassies, legations, consular offices, and other facilities overseas, including all military facilities and naval vessels and stations. Chuck Schumer's spokesperson, Schumer and Senate Majority Leader Mitch McConnell also requested the Department of Defense to "provide necessary support so that US flags on all government buildings remain at half mast through sunset" on the day of McCain's interment.
After departing the state Capitol, McCain will be honored in a memorial service celebrating his life and legacy at the North Phoenix Baptist Church.