When President Obama greets African-Americans who broke barriers, he almost invariably uses the same line.

“I wouldn’t be here if it weren’t for you,” he said to Ruby Bridges Hall, who was the first black child to integrate an elementary school in the South. The president repeated the message to a group of Tuskegee airmen, the first black aviators in the United States military; the Memphis sanitation workers the Rev. Dr. Martin Luther King Jr. addressed in his final speech; and others who came to pay tribute to Mr. Obama and found him saluting them instead.