The 30-year-old suffered the problem during his side's absorbing 22-16 defeat at the hands of France on Sunday.
The South African-born forward will now return to his club Glasgow Warriors to undergo treatment on the injury and will have a review of his progress in approximately six weeks' time.
"Scotland number eight Josh Strauss will take no further part in the RBS 6 Nations Championship after sustaining an injury to his kidney," read a statement on the Scottish Rugby website.
"The 30-year-old suffered a blow to his flank in the France vs. Scotland RBS 6 Nations match on Sunday, 12 February.
"After completing the match, he was scanned in Paris. Both the scan, and the player, were reviewed at Spire Murrayfield Hospital back in Scotland, where the full extent of the injury was confirmed."