Without mentioning the president by name, Bush suggested it was out of step with the country's values to have driven protesters from Lafayette Square across from the White House on Monday just before Trump walked through for a photo opportunity.

"The only way to see ourselves in a true light is to listen to the voices of so many who are hurting and grieving," Bush said in a statement on Tuesday.

"Those who set out to silence those voices do not understand the meaning of America -- or how it becomes a better place."

Trump later issued a Twitter post lauding authorities for using "overwhelming force" and "domination" in Washington.

The Washington Post reported on Tuesday that the order to remove the protesters came from Attorney General William Barr.

With demonstrations, sometimes marred by violence, erupting across the country since Floyd died on a Minneapolis street on May 25 with a police officer's knee on his neck, Bush said he and his wife, Laura, were anguished by "the brutal suffocation."

The latest incident of an unarmed black man dying at the hands of a white police officer raises troubling questions that need to be confronted, he said.

"It is time for America to examine our tragic failures - and as we do, we will also see some of our redeeming strengths," said the 43rd US president, who served from 2001 to 2009.

"It remains a shocking failure that many African Americans, especially young African American men, are harassed and threatened in their own country," he said.

"This tragedy -- in a long series of similar tragedies -- raises a long overdue question: How do we end systemic racism in our society?"

Australian Associated Press