The Leinster stalwart was withdrawn after suffering the blow during a 24-12 defeat to England at Twickenham on Sunday
Healy has been ruled out of the clash with Italy at the Aviva Stadium on March 7 and will not face leaders France in Paris on the final day of the tournament.
Pro14 champions 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.