Postecoglou led Spurs to the Europa League title this season - a first UEFA trophy in 45 years won by the club. The triumph also sees Tottenham handed a place in next season's Champions League.
But Tottenham's 17th place in the Premier League has cost Postecoglou his job, despite the squad's injury crisis throughout the campaign.
Vicario posted to social media today: "Boss, I just want to say a massive thank you for everything you have done for me and for all of us.
"From that very first call, right from the beginning, you always showed so much belief in me.
"Giving me the opportunity to be part of the leadership group... those moments, and many others, will stay with me forever."
Incredible person
Vicario continued: "You are not only a top manager, you are an incredible person to work for, a real leader, a mentor, and someone I'll always look up to.
"What we achieved TOGETHER will stay in the history books.
"Wishing you nothing but success as I know you will go on to achieve more and more.
"Thank you, Boss. Forever grateful, Vic."