Callahan replaces the retired Chris Drury, who had been captain since the beginning of the 2008-09 season.
"Ryan Callahan embodies all the leadership qualities we seek from our captain," said New York president and general manager Glen Sather in a statement. "He leads by example with courage and a tireless work ethic on and off the ice, which is why he is so deserving of this honour."
Callahan, 26, has been a Rangers alternate captain for the past two seasons. He is entering his fifth NHL campaign and has 76 goals and 68 assists in 284 regular season games.
Richards, 31, was signed as an unrestricted free agent by the Rangers on July 2. Staal, 24, was a first-round pick of the Rangers in 2005 and has played four seasons in New York.