Real manager Zinedine Zidane had played down concerns about Bale's fitness, saying he rested him for Sunday's win over Espanyol as a precaution.
Bale joined the Wales squad on Monday and, despite sitting out training, it had been hoped he would be fit.
But the 28-year-old's injury means he will not travel to Georgia.
Barnsley striker Tom Bradshaw has been called up to Wales the squad in his place.
Bale's absence in Tbilisi on Friday and in Cardiff for the visit of the Republic of Ireland three days later is a huge blow to Welsh hopes of qualifying for a first World Cup since 1958.
Wales are second in Group D, trailing leaders Serbia by four points with two matches remaining.
Second place could be enough to earn Chris Coleman's team a place in the play-offs, which will see eight of the nine runners-up in qualifying groups play for four remaining places in Russia 2018.
In order to secure a play-off place, Wales would realistically need at least four points from the games against Georgia and the Republic of Ireland.
Their chances of doing so will be harmed by the injury to Bale, who has scored four goals so far in this campaign. Wales have failed to win a match without the Real Madrid forward since 2013, drawing three and losing three matches in his absence.
The former Tottenham player has also been in fine form for Real, scoring a brilliant goal in last week's 3-1 Champions League win over Borussia Dortmund before being forced off with the muscular problem which has sidelined him for this month's qualifiers.
Bale was in Cardiff on Monday to receive his players' player of the year award at the Football Association of Wales' annual ceremony.
BBC pundit and ex-Wales international Robbie Savage
For Wales to be without Gareth Bale for two massive games is devastating.
His passion and desire to play for Wales is there for all to see... it is a huge, huge blow.
It will all be on the last game with Ireland, I believe. A point in Georgia won't be a bad result for me now.
But even if Wales win their two games, that might not be enough. These are two of the biggest games in Welsh football history.
Fingers crossed the lads can go out there and perform, but it is going to be very, very difficult.