In Falcon Heights, the suburb where Mr. Castile was shot, a group of university students and employees marched to the shooting scene to pay tribute Friday afternoon. And at the governor’s residence in a residential area of St. Paul, demonstrators continued to mingle outside the gates. The protests have been large, spirited and almost entirely peaceful, though one person was arrested and a police car was damaged early Friday near Gov. Mark Dayton’s home when demonstrations turned tense.

So far, Justice Department officials have said they were monitoring the state investigation, but have not announced their own inquiry. John J. Choi, the prosecutor in Ramsey County, said he had urged the state agency investigating to be prompt and thorough, but did not offer a timeline on when a charging decision might be made.

Mr. Choi, whose office will decide whether to bring charges, said he was unsure whether he would present the evidence to a grand jury or make the charging decision himself. Mr. Choi historically has used grand juries in police shooting cases and said he saw benefits in doing so, but that he would consider whether that was the right approach for Mr. Castile’s case.

“I just need a little time and thought put into it,” Mr. Choi said at a news conference at his downtown office. “I think this is a very extraordinary case.”

If Mr. Choi opts to decide on charges himself, rather than presenting the case to grand jurors, he would be following his colleague in neighboring Hennepin County, which includes Minneapolis. The prosecutor there, who also had long used grand juries for police shootings, changed his policy and decided himself to not charge the Minneapolis officers involved in last year’s fatal shooting of Jamar Clark, another case that prompted widespread protests in this area.

Governor Dayton, a Democrat, who angered some in law enforcement Thursday by saying he thought Mr. Castile probably would not have been shot if he were white, said Friday that he was standing by those comments.

Mr. Dayton, who has at times mingled with the protesters outside his home, said he appreciated the demonstrators’ peaceful tone. On Friday, he said he planned to meet with African-American pastors and civic leaders in the coming days, and called for people to “react nonviolently.”

“I make an appeal to everyone in Minnesota for calm, for understanding of this difficult time and the need for calm and nonviolence,” said Mr. Dayton, who also spoke of a shooting in Minneapolis on Friday that left a toddler dead. “We’ll get through this terrible time here in Minnesota if we can all recognize that and not take any actions that are going to exacerbate a very difficult situation.”