Ardiles' former Tottenham team-mate Ricky Villa was also involved in the incident on Monday, which took place while the pair were on the island to film a documentary.
A spokesperson for the Falkland Islands Government revealed Ardiles had now travelled home to Argentina to continue his treatment.
She said: "I can confirm that (soundman) Carlos Marcellan and Ossie Ardiles were last night transferred to Buenos Aires for further investigations. We hope they both have a speedy recovery."
Ardiles was taken to King Edward VII Memorial Hospital in Stanley following the accident, and had passed on his appreciation to all those who assisted him and the others injured.
"Thank you to everyone in the Islands who has helped, without exception," he said.
"From the moment of the accident, the attention and care we have received from the police, the RAF helicopter team, the doctors and nurses here has been world class.
"Thank you to everyone for your concern, I am fine and doing well."