The 36-year-old Victorian, who sustained the injury during last month's Indian tour by the Australian 'A' team, missed the ongoing four-day match against the Board President's XI here.
Australian team physiotherapist Alex Kountouris said the decision to send McGain home was made after it was confirmed he would not be able to bowl for the majority of the tour.
"Bryce suffered a setback at training on Tuesday in an attempt to be available for the four-day match," Kountouris said in a Cricket Australia statement.
"As a result, it has been decided that Bryce will not be available for selection for the majority of the tour."
A decision on who will replace McGain for the four-Test series, which starts next week, will be taken after Australian chairman of selectors Andrew Hilditch arrives here early Saturday.
It is expected that left-armer Beau Casson, a surprise exclusion from the original squad after making his Test debut in the Caribbean earlier this year, will be the likely replacement.
McGain was widely tipped to make his Test debut as the first choice spinner ahead of fellow newcomer Jason Krejza, who went for 0-123 in the practice match earlier on Friday.
The world champions are struggling to throw up a quality spinner following the retirements of legendary Shane Warne and fellow wrist spinner Stuart McGill in quick succession over the last two years.
The series for the Border Gavaskar Trophy opens in Bangalore next Thursday, followed by back-to-back Tests in Mohali, New Delhi and Nagpur.