The Leinster stalwart was withdrawn after suffering the blow during a 24-12 defeat to England at Twickenham.
Healy has been ruled out of the clash with Italy at the Aviva Stadium on 7 March and will not face France in Paris on the final day of the tournament.
Pro14 champion Leinster will be hoping the 32 year-old is available for a European Champions Cup quarter-final against holders Saracens in Dublin on April 4.
Healy will have to wait to bring up a century of caps, with his appearance against England taking him to 98.
Will Connors, Max Deegan, Chris Farrell, David Kearney, Luke McGrath and Jack O'Donoghue were on Tuesday released from international duty to play for their provinces this weekend.
Defeat to the Rugby World Cup runner-up last weekend ended Ireland's hopes of winning a Grand Slam and leaves it level with England, four points adrift of France.