The former Russian spy had been in intensive care since he and his daughter Yulia were poisoned with the nerve agent novichok on 4 March.
His improvement was announced following reports that Yulia had told her cousin that her father was "getting better".
Dr Christine Blanshard, medical director at Salisbury District Hospital, said on Friday: "Following intense media coverage yesterday, I would like to take the opportunity to update you on the condition of the two remaining patients being treated at Salisbury District Hospital.
"Last Thursday, I informed you that Yulia Skripal's condition had improved to stable. As Yulia herself says, her strength is growing daily and she can look forward to the day when she is well enough to leave the hospital.
"Any speculation on when that date will be is just that - speculation. In the meantime, Yulia has asked for privacy while she continues to get better - something I'd like to urge the media to respect.
"I also want to update you on the condition of her father, Sergei Skripal. He is responding well to treatment, improving rapidly and is no longer in a critical condition.
"As you'll appreciate, I won't be giving any further updates at this time."
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