The 26-year-old flanker has been hampered by the injury for a few months and it has been decided the best course of action for all concerned was for him to go under the knife immediately.
Edinburgh boss Richard Cockerill said they could have tried to continue to manage the situation but after consultation with Scotland's medical staff and head coach Gregor Townsend it was deemed necessary for the open-side to seek treatment as soon as possible.
Cockerill said: "It's something we probably could have managed through, but he's been struggling with for a little while. We've chosen to get it done now because he'll then be fully fit for pre-season and he can have a summer off and recuperate.
"He's a relatively small man in the modern game who plays big and it's a World Cup year.
"So, in consultation with Gregor Townsend, the right thing to do for the player is to get that done now, rather than play at 80 per cent or 90 per cent."
Watson has 20 caps and featured in all of his country's recent games in the Six Nations but will not travel as part of the tour heading to Canada, USA and Argentina in June.