De Vries has been replaced at the Faenza-based operation by Daniel Ricciardo for the remainder of the 2023 season.
After catching the eye in a cameo for Williams at last year's Italian Grand Prix, the Dutchman failed to impress Red Bull bosses in his 10 races with AlphaTauri.
Pressure had been mounting on de Vries, with the Formula E champion staving off speculation as far back as the Monaco Grand Prix.
With his fate confirmed last week, he had kept his counsel until posting a short note on social media.
“After recent events, I decided to take some time for myself away from social media, which I will continue to do,” wrote de Vries.
“I would like to thank Red Bull and Scuderia AlphaTauri for the opportunity to live my dream.
“Of course, it hurts that the F1 chance I dreamed of for so long ended prematurely.
“But life is not a destination, it's a journey, and sometimes you have to take the hard road to get where you want to be.
“I am grateful for our privileged lives, proud of our journey and my family.
“This is just another experience, we move on and look forward to the next chapter,” he added.
“Thank you to everyone for your kind and encouraging messages in the past week. It's been heart-warming to feel your support.”
Ricciardo steps into the seat after working as Red Bull's third driver since the start of the year.
Ricciardo is due to face the media, including Speedcafe, in Hungary today.