Albon was replaced in the driver lineup ahead of the weekend's Italian Grand Prix, with Dutch driver Nyck de Vries making his Formula One race debut at Monza and finishing ninth.
Albon had successful laparoscopic surgery on Saturday before he encountered respiratory complications.
"Following surgery, Alex suffered with unexpected post-operative anaesthetic complications which led to respiratory failure, a known but uncommon complication," Williams said in a statement.
"He was re-intubated and transferred to intensive care for support. He made excellent progress overnight and was able to be removed from mechanical ventilation yesterday morning.
"He has now been transferred to a general ward and is expected to return home tomorrow. There were no other complications."
Williams added Albon's focus is now on preparing for the Singapore Grand Prix over the Sept. 30-Oct. 2 weekend.
