Sala, 28, disappeared on 21 January with pilot David Ibbotson during a flight between Nantes and Cardiff.
Sala had been in the French city to say goodbye to his former teammates following his 17 million euro transfer from the French Ligue 1 club to the English Premier League side Cardiff City.
"We can now begin to mourn our son and our brother," the Argentine's family said in a statement.
"We would like to thank you for all of your signs of affection and support in the most painful time of our life. To see everyone doing everything possible to join us in the search has been a precious help to us."
The family also sent its condolences to the relatives of 59-year-old Ibbotson whose body has yet to be recovered.
"We hope the authorities do everything possible to find him," added the Sala family statement.
Sala had been in France since 2010. He launched his career with the Bordeaux youth teams before stepping up into the club's senior squad. A series of loans took him to Orléans and Caen before he signed for Nantes in 2015.
In tribute to the player, Nantes on Friday said it would withdraw his number 9 shirt.
"I'm devastated," said the Nantes president Waldemar Kita in a statement. "I don't know what to say. Emiliano left his mark on us all and that's why I and many supporters want to honour him by retiring his shirt. He will be among the legends of the club."
French football chiefs on Friday announced all top flight and second division games would be preceded by a minute of applause for Sala.