The NHL announced the ban after holding an in-person hearing with Gillies at the league's office in Toronto on Friday.
The suspension is Gillies' second of the season, and comes almost immediately after he finished serving a nine-game ban for his role in a brawl against the Penguins on Feb. 11. Gillies' incident against the Wild this week came during his first game back after that punishment.
On Wednesday, Gillies slammed Clutterbuck into the boards in retaliation for a hit that Clutterbuck put on New York's Justin DiBenedetto moments earlier in the opening minutes of the second period. Gillies received a five-minute major for hitting from behind, plus 10-minute and game misconducts for his actions.
"By targeting his opponent's head, three shifts into his first game back from a suspension for a very similar action, Mr. Gillies has forfeited his privilege of playing in the league for 10 games," said NHL Senior Executive Vice President of Hockey Operations Colin Campbell in a statement. "While it is fortunate there was no injury on the play, there can be no justification for a player delivering a dangerous check to an opponent in this manner".
The league also noted in its release that Gillies is now considered a repeat offender under terms of the Collective Bargaining Agreement.
The 32-year-old has one goal and 124 penalty minutes in 33 games this season. He will be allowed to return to New York's lineup on March 26 against Philadelphia.