Felipe Massa's condition is "severe but not critical" after surgery on his fractured skull, but there is concern about damage to his left eye.
The Ferrari driver's surgeon confirmed Massa's eye was injured when a spring from Ruben Barrichello's car hit him in qualifying in Hungary on Saturday.
"We don't know if he'll be able to race again. It's too early to say about his future," added surgeon Robert Veres.
"But the immediate life-threatening condition has been averted."
Massa has been taken out of an induced coma and is able to communicate "actively" and move his hands and legs but cannot speak.
A Ferrari spokesman said: "Felipe had a quiet night. He is OK and he is due to have another CT scan."
A Hungarian hospital spokesman revealed they are "optimistic" that a "slow recovery is beginning".
Massa's engineer reflects on crash (UK only)
"He's woken up [from sedation] more often and is able to communicate actively," added Istvan Bocskai, a spokesman for the Hungarian defence ministry, which administers the Budapest hospital where Massa is being treated.
Ferrari president Luca di Montezemolo flew to the AEK military hospital on Monday to be at Massa's bedside, alongside the driver's father, mother and pregnant wife, who have all flown in from Brazil.
Di Montezemelo staged a joint news conference with Massa's doctors and said the team were focused on their driver's recovery and not on considering any possible replacements.
"For us, the first priority is to find out Felipe's recovery progress and situation. He is a very important member of the Ferrari family not just the Ferrari team," said Di Montezemelo.
"First we will find out the situation with Felipe and then we will see and we will think, without pressure.
"Only at that moment will we make a decision and if we have to take a decision we will make a good decision."
Massa was injured when a spring from fellow Brazilian Barrichello's car struck his helmet during qualifying for the Hungarian Grand Prix, with the Ferrari driver then hitting a tyre wall.
Following Kimi Raikkonen's second-place finish at the Hungaroring on Sunday, Ferrari team principal Stefano Domenicali dedicated the result to Massa.
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"At this time all our thoughts are with him and his family and I think this sentiment also applies to those who are close to him and to fans all over the world," stated Domenicali.
"We got the best result we were capable of but please allow me on behalf of everyone at Ferrari to send our best wishes to Felipe. We love you and we are all thinking of you."
Race winner Lewis Hamilton extended his goodwill to Massa following the Briton's first victory of the season.
"Maybe I can speak for all the drivers but Saturday was quite a sad day and we missed Felipe," commented the defending world champion.
"I'm just glad the surgery went well and we're just going to keep him in our thoughts and prayers, and we wish him a speedy recovery.
"He's got a child coming, so he needs to get back into top shape so when he's finally a dad he can do the job well."
After being struck on the helmet by the 1kg spring from Barrichello's car, Massa's feet are believed to have come to rest on both the throttle and brake and his car ploughed head-on into a tyre wall at about 125mph.
Barrichello reassured by Massa's doctors
Massa was swiftly attended to by trackside medics before being taken to the circuit's medical centre, after which he was airlifted to hospital.
Barrichello revealed his thoughts were with Massa, even during the race.
He said: "I would love to see him because he is a friend, not just because the spring came from my car.
"It is for the fact that I know he would have done the same for me. But at the same time I have to think of my kids, of my family, and to be there for them, so it is a bit of an emotional moment.
"It is burning a little bit in my stomach and I would be lying if I said I wasn't thinking of him in the the middle of the race."
Massa's accident came days after Formula Two driver Henry Surtees, 18, was killed in what was described as a freak accident during a race at Brands Hatch.
The son of motorsport legend John Surtees was struck by a wheel that flew off a competitor's car.
Barrichello has led the calls for more stringent safety measures to be looked into. Referring to Surtees, the veteran refused to believe the two incidents were coincidental..
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"I honestly don't believe in coincidences in life," he said on Saturday. "Things happen for a reason and I think this is the second message.
"Imola [where Ayrton Senna and Roland Ratzenberger died in 1994] was a message and the cars were improved. Unfortunately, we lost a boy [Surtees], which is tremendously sad.
"It is not a coincidence something happened right now. In the Grand Prix Drivers' Association we talked quite a lot about it yesterday - and something needs to be done. Absolutely."
Button and Barrichello's team boss Ross Brawn said the team are still investigating why the spring came loose from the back of their car.
"We haven't had the piece back from [motorsport governing body] the FIA yet so once we get it back we can understand what occurred," he said.
"It's the first time we've really had a problem with the car as it has been so reliable."