The 34-year-old described Carrick's managerial situation as a "big problem" for the club's hierarchy because of the rapid transformation he has overseen in recent months.
Casemiro, who is expected to join Inter Miami CF when his Manchester United contract expires this off-season, said he had complete faith in Carrick's ability to lead the club long term.
The Brazilian also agreed with comparisons between Carrick and legendary former United manager Sir Alex Ferguson.
According to Casemiro, Carrick has demonstrated a calm authority and leadership style that separates him from other candidates for the role.
The midfielder added that the improvements made under the interim manager had been immediate and visible throughout the squad.
Casemiro himself has enjoyed a resurgence in form since Carrick took over, although he was absent for United's goalless draw against Sunderland on Saturday. Carrick has remained in interim charge since January while the club continues its search for a permanent manager.
"He deserves to stay," Casemiro said on the Rio Ferdinand Presents YouTube channel.
"The team plays very good. He will win any games. The team improves day by day. Game by game the team improves.
"So for me it's a big problem because in my opinion he deserved this opportunity. So, especially for me, because he played in the midfield, he talked with me as a midfielder to a midfielder, it's easier.
"But for me, he deserved the opportunity because the team is very comfortable. The team plays very good. We have big wins. So for me he deserved this opportunity."
