Voges was forced to retire hurt on 86 after suffering the injury while batting during the County Championship draw with Somerset on 4 June.
Middlesex hope to have the Australian back for their T20 Blast game against Hampshire at Lord's on 3 August.
The 37-year-old has scored 230 runs in four first-class games this season at an average of 57.50.
"Losing Adam is naturally a blow as he contributes significantly to everything we do," director of cricket Angus Fraser said.
"His absence, however, does give opportunities to other players who are desperate to play in the first XI.
"In the past we have coped well without an overseas player and I expect us to do so now."
Defending county champions Middlesex are sixth in the Division One table, having drawn all four of their matches this season.
Voges retired from international cricket in February with a Test batting average of 61.87 from 31 innings, second only to Australian great Sir Donald Bradman's career average of 99.94 from 52 Tests.

The former Ireland head coach, 53, had endured a difficult time since taking charge in April last year.
A year ago, he was suspended and then reinstated following comments he made about team selection.
Joel Garner, Henderson Springer and Roddy Estwick will lead the team when they face Pakistan in three T20s, three ODIs and three Tests in the UAE.
The first match - a T20 game - takes place on 23 September in Dubai.
The West Indies Cricket Board (WICB) cited "differences in culture and strategic approach" with Simmons.
The former all-rounder had questioned the omission of all-rounders Dwayne Bravo and Kieron Pollard from the one-day squad last year.
He was suspended by the WICB, who said Simmons had appeared to "question the legitimacy of the selection process".
The departure of Simmons follows the sacking of Curtly Ambrose as bowling coach and Darren Sammy as T20 captain.
West Indies are currently third in the world T20 rankings but lie eighth in both Tests and one-day internationals respectively.

Oliver Park, 51, ignored safety signs and warnings from wardens and went into a restricted area close to the edge of a cliff on Wednesday, officials say.
They say he tried to pose for a photograph of himself leaping as if in flight but lost his footing.
Rescuers found the body on Thursday and took it to a local police station.
Police said his body would be taken to a morgue in the city of Cusco.

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The victory means the Irish will face France in Sunday's final at the World League 2 tournament.
Wales will also make the final qualifiers in Johannesburg if they beat Scotland in the third place play-off.
Gareth Furlong put Wales ahead before Shane O'Donoghue, Ben Walker and Neal Glassey netted for the Irish.
Ireland, 10th in the world rankings, were strong favourites to account for the 34th-ranked Welsh but the visitors produced a strong challenge.
Furlong's early goal from a penalty corner left Wales ahead after the opening quarter but O'Donoghue and Walker struck in quick succession in the second quarter to give the hosts a half-time lead.
Wales had a couple of chances in the third quarter - including a missed penalty corner - before Glassey sealed Ireland's win by netting early in the final quarter.
France earned a 2-0 win over the Scots in Saturday's second semi-final.

Page, 20, has agreed a two-and-a-half-year contract at The Valley.
He made three first-team appearances for West Ham, all in the Europa League, and has previously been on loan at Coventry City and Cambridge United.
"He's quick, likes to get forward and is a good crosser. I really like that in a full-back," Charlton manager Karl Robinson told the club website.
"I got a chance to see him during his loan spell at Coventry and he's a left-back that fits in well with our system.
"We're building a squad here to get into that top six and Lewis will be an important part of what we are trying to do."
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Middlesex batsman Adam Voges will be out until August after suffering a torn calf muscle in his right leg.
[NEXT_CONCEPT]
Phil Simmons has been sacked as West Indies coach just five months after their stunning World T20 triumph.
[NEXT_CONCEPT]
The body of a German tourist has been recovered from a ravine after he fell while visiting the ancient site of Machu Picchu in the Peruvian Andes.
[NEXT_CONCEPT]
Ireland's men guaranteed themselves a spot at the final World Cup qualifying tournament in the summer by beating Wales 3-1 at Stormont on Saturday.
[NEXT_CONCEPT]
League One club Charlton have signed left-back Lewis Page from West Ham for an undisclosed fee.