The Dutchman, who has inked a contract to June 2027, replaces the outgoing Steve Cooper after his sacking last Sunday.
Van Nistelrooy joins Leicester with the club 16th in the Premier League and just one point above the relegation zone.
"I’m proud, I’m excited," he told LCFC.com. "Everybody that I speak to about Leicester City Football Club is enthusiastic. They have great stories about the quality of the people working at the Club, the supporters and, of course, the recent history of the Club is impressive.
"I’m excited to start and to get to know everyone and give everything I can for the Football Club."
Van Nistelrooy's appointment comes just weeks after leaving his position as Manchester United's interim manager following Erik ten Hag's exit.
The 48-year-old won three games and drew one during his time in charge, with two of those victories coming against Leicester.
However, he left the club after being told by incoming United boss Ruben Amorim that he would not have a role among the backroom staff.
Leicester City chairman Aiyawatt Srivaddhanaprabha said: "It is my pleasure to welcome Ruud to Leicester City. He joins a Club with a rich history, passionate supporters, and a talented squad, and we are all excited to see the impact he can have as we embark on this new chapter together.
"Ruud’s experience, knowledge, and winning mentality will undoubtedly bring great value to us, and we look forward to supporting him in achieving success for our fans and our Club."
Leicester will be Van Nistelrooy's second full-time managerial position after his role with PSV Eindhoven.
There, he won the Dutch Cup in the 2022/23 campaign but quit before the end of the season after citing a lack of support from the club.
Van Nistelrooy takes the place of Cooper, who was dismissed by Leicester following their recent 2-1 defeat at home to Enzo Maresca's Chelsea.
He will take charge of the team after their match against Brentford on Saturday, with his first game on Tuesday evening against West Ham.
Cooper, 44, was appointed by the Foxes in the summer and won just two of his 12 league games.