The 25-year-old suffered the injury during the 39-16 victory over the United States on Sunday which put the Scots top of Pool B.
“He's worked so hard to get fit. When we started off in June he was 12 months behind everyone else, so for him to get where he got to and playing as well was an achievement in itself. To get shot down like that, we all feel for him," Scotland assistant coach Duncan Hodge told reporters.
Gilchrist broke his arm playing for his club Edinburgh in October last year having just been made Scotland captain by coach Vern Cotter. He missed the rest of the season.
Flyhalf Finn Russell sustained an ankle knock that will require further care and assessment.
Russell is on crutches but Hodge said that was just a "medical precaution".
"With all these soft-tissue injuries, it’s a wait and see.
"Obviously we don't want to lose anyone and 10 is a key role. Duncan (Weir) did well when he came on and he's a great 10 as well, so there is no undue panic from us there.”
Scotland next play South Africa in Newcastle on Saturday.
(Reporting by Justin Palmer; editing by Ed Osmond)
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