The injury also casts doubt on McKay's inclusion in the Scotland squad for the Elite Stage of the European Under-19 Championship towards the end of May.
Another Rangers player, Lewis Macleod was already battling a knee injury to make the squad.
McKay came off after 16 minutes of the 2-0 friendly victory and will see a specialist in London early next week to gauge the extent of the damage.
But Rangers said it "seems certain" that the injury will rule him out of Rangers' four remaining Irn-Bru Third Division matches.
Meanwhile, the club revealed that defender Ross Perry had suffered a setback in his bid to recover from a hamstring injury.