WASHINGTON — As he signed a proclamation marking the holiday next week honoring the Rev. Dr. Martin Luther King Jr., President Trump on Friday recalled the civil rights leader’s message that “no matter what the color of our skin or the place of our birth, we are created equal by God.”

Then came the questions from reporters.

“Mr. President, are you a racist?”

He did not answer and instead headed for the door.

It is a question many were asking after the latest charged episode in a presidency that has played out along the nation’s racial fault lines from its beginning. Mr. Trump’s comment to lawmakers that the United States should accept more immigrants from places like Norway instead of from Haiti or “shithole countries” in Africa did not sound consistent with the notion that all people are equal no matter the place of their birth or the color of their skin.

If it were a one-time comment, an inadvertent insensitivity, it would still have stirred a firestorm. But Mr. Trump has said so many things on so many occasions that have rubbed the raw edges of race in America that they have raised the larger issue. A country tainted at its founding by slavery and struggling with that legacy ever since is now led by a chief executive who, intentionally or not, has fanned, rather than doused, the fires that divide white, black and brown.