“Would this have happened if the driver were white, if the passengers were white?” Gov. Mark Dayton asked at a news conference. “I don’t think it would have.”

He said he had heard from many black people, including some in positions of authority, about “how they’ve been pulled over, singled out,” in a way that white people would not have. “I’ve been told by very respectable African-American leaders that they understand how this dynamic goes on.”

Mr. Dayton and members of the state’s congressional delegation asked for the Justice Department to investigate the death of Philando Castile, 32, who died hours after the department took over the investigation into the fatal police shooting, also captured on video, of another black man in Baton Rouge, La. The governor said he had spoken with Denis McDonough, the White House chief of staff; Valerie Jarrett, senior adviser to President Obama; and Vanita Gupta, who heads the Justice Department’s civil rights division.