The Scotland international prop will now begin a period of rehab and is expected to remain on the sidelines for up to six months.
Meanwhile, Dougie Hall has had a succesful operation on his arm.
The hooker was substituted in the early stages of the RaboDirect PRO12 Final and will be out of action for several months.
Glasgow Warriors head coach Gregor Townsend said: "It's disappointing for Ryan (Grant) and Dougie (Hall).
"They've both played a lot of rugby this season and are key members of our squad.
"We wish them all the best with their rehab."
Guinness PRO12
Suite 208, Alexandra House,
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Ballsbridge, Dublin 4, Ireland