The letter also accuses Fox of violating an agreement not to leak Carlson's private communications to the media and not to use his private messages against him.
Gamble was the NBCUniversal employee who filed a sexual harassment complaint that led to the eventual ouster of the company's CEO Jeff Shell last month.
Borrell Associates projects that figure will climb considerably over the next four years as the industry undergoes what it describes as a "great reshuffling."