Healy will sit out the February 24 trip to Murrayfield and the March 9 home tie with the French having been told he won't be free to play until the day after the Round 4 encounter at the Aviva Stadium.
The 25-year-old appeared in front of a disciplinary committee in London on Wednesday after escaping a yellow or red card for the first-half altercation in Dublin.
The disciplinary panel categorised Healy's actions as a mid-range offence, with that level of seriousness carrying a minimum five-week sanction.
That was reduced by the maximum of two weeks for mitigation, leaving Healy to face three weeks and two Tests on the sidelines.
Healy, who is ironically being tipped to start alongside Cole in the Lions' front row this summer, will be available for selection for Ireland's final tournament fixture with Italy in Rome on March 16, although he may also still appeal the suspension.
His immediate absence is likely to open the door to a return to the matchday 23 for Ulster loosehead Tom Court, who missed the opening two fixtures after being overlooked by head coach Declan Kidney.
Court is expected to be named on the bench for the Scotland game, with Munster's Dave Kilcoyne in line for a promotion to the starting XV having been on the bench in Rounds 1 and 2.
Healy's suspension is another worrying blow for Kidney, who has already lost Simon Zebo and Jonathan Sexton to injury this week and has concerns over the likes of Brian O'Driscoll, Donnacha Ryan and Mike McCarthy.
In-form wing Zebo has been ruled out of all three remaining Championship matches with a broken foot, while star playmaker Sexton is out for at least a fortnight with a hamstring problem.
Irelandare already without Lions trio Paul O'Connell, Tommy Bowe and Stephen Ferris, although Ferris is hopeful of returning to domestic action with Ulster early next month.