GOSH referred the case back to the High Court after reports of "new" data from foreign healthcare experts suggested treatment could improve his condition.
Charlie's parents have made several unsuccessful challenges to a decision to end the 11-month-old's life support.
GOSH told the hearing the evidence was not new but it was right to explore it.
Mr Justice Francis is overseeing the preliminary hearing in the Family Division of the High Court.
In April he ruled that Charlie's life support should be ended and said earlier it would take something "dramatic and new" to make him change his mind.
Mr Justice Francis is due to resume hearing the case on Thursday.
On Sunday, Charlie's parents Chris Gard and Connie Yates handed in a 350,000-signature petition calling for him to travel to the US for treatment.
Charlie's parents are seeking permission to remove their son - who has the rare condition mitochondrial depletion syndrome - from the care of Great Ormond Street Hospital so he can undergo experimental treatment abroad.
Charlie's medical case explained
The judge said: "There is not a person alive who would not want to save Charlie."
A lawyer for the family said new and unpublished data was recently shared with the hospital that suggested treatment could produce a "dramatic clinical improvement" in Charlie's condition.
Previously doctors had indicated Charlie had irreversible structural brain damage.
Lawyers representing the family have now said using "cutting edge genetic science" there was a "small chance" of brain recovery and that it was a chance "worth taking".
They questioned whether Mr Justice Francis was the correct person to assess the latest medical evidence, given that in April he had ruled Charlie's life support should be withdrawn.
In reply, the judge said: "I did my job.  I will continue to do my job."
At the hearing, a lawyer for GOSH said the alleged "new research" had been available for the judge's consideration in April and was purely lab-based anyway, and related to patients with muscle problems only rather than brain damage.
Charlie inherited the faulty RRM2B gene, affecting the cells responsible for energy production and respiration and leaving him unable to move or breathe without a ventilator.
GOSH describes proposed experimental therapies as "unjustified" and said the treatments being offered are not a cure.
However, the hospital's decision to go back to court came after researchers at two international healthcare facilities said they had "fresh evidence about their proposed experimental treatment".
Charlie's parents, from Bedfont in west London, want their son to have nucleoside therapy.
Speaking to BBC Breakfast, Ms Yates described the situation as a "living hell".
"I couldn't sit there and watch him in pain and suffering, I promise you I wouldn't," she said, adding: "I think parents know when their children are ready to go and they've given up, and Charlie is still fighting.
"It's horrible that this decision has been taken out of our hands. It's not just about us knowing best, it's about having other hospitals and doctors saying we want to treat [Charlie] and we think it's the best thing to do."
Ms Yates said they were not criticising Great Ormond Street Hospital as "they do great things".
She said she hoped the judge would take into account new evidence as when the decision was made previously, Charlie's chance was rated at being close to 0% but now this has increased to 10%.
Charlie is thought to be one of 16 children in the world to have mitochondrial depletion syndrome.
It is a rare genetic condition which causes progressive muscle weakness and brain damage because he is unable to get energy to his organs.
Doctors have said he now cannot see, hear, move, cry or swallow and has irreversible brain damage. His lungs are only able to keep going because of the treatment he is receiving.
They have argued he should be allowed to die with dignity.
But his parents and supporters have been fighting for him to be given an experimental treatment in the US.
The treatment is not a cure - there isn't one - but it has been suggested it could reduce the effects of the disease.
Although doctors in the US have since said the benefits they have seen have not been in cases as advanced as Charlie's.
US President Donald Trump and the Vatican have supported the parents' campaign for Charlie to be treated abroad, but a leading expert has described interventions from high-profile figures as "unhelpful".
Prof Neena Modi, president of the Royal College of Paediatrics and Child Health, said in an open letter that Charlie's situation was "heartbreaking" for his parents, but added that even well-meaning interventions from outsiders could be unhelpful.
Mr Gard said: "If we won the court case and we got to America, and then within the first week of treatment he started suffering and he was in pain, we would let him go.
"This isn't about us. This is about Charlie and giving him the chance he needs."
Get news from the BBC in your inbox, each weekday morning
A lawyer for Great Ormond Street Hospital (GOSH) has dismissed claims of "fresh" medical evidence in the case of terminally ill baby Charlie Gard.