Buyers from around the world bid for the textile, which was used at the coronation of Edward VII, when it went under the hammer at Lyon & Turnbull. It had been valued at £15,000 but sold for £79,250.
It was gifted to the Cathedral's Kirk Session upon the death of Charles Hepburn, who bought the carpet in 1938.
The Church of Scotland said it was advised its charity status would not allow funds to be spent on conserving the carpet.
Dr Scott Hepburn, Mr Hepburn's nephew, has accused the Church of "selling off the family silver".