Speaking outside the hospital, Charlie's parents, Connie Yates and Chris Gard thanked their supporters and the media for sharing the story of their 11-month-old son worldwide.
But they lost a lengthy legal battle after successive judges ruled in favour of doctors at GOSH, who argued the treatment would not improve the infant's quality of life and say his life-support machine should be switched off.
Unless the court hearing produces a change, the hospital is barred by a series of court decisions from allowing the baby to be taken elsewhere for treatment.
Charlie was born in August with mitochondrial DNA depletion syndrome, a progressive disease that causes muscle weakness and loss of motor skills, leaving those who have it unable to stand, walk, eat, talk and eventually breathe. "We feel that it should be our right as parents to decide to give him a chance at life", Yates, carrying a petition signed by some 350,000 people supporting the couple's quest, said.
Asked if it was right that judges could overrule the wishes of Charlie's parents, Mr Lidington told Sky News's Ridge On Sunday: "It is right that judges interpret the law, independently and dispassionately".
That prompted the hospital to seek another High Court ruling.
Chris Gard told reporters Sunday that the USA hospital where they want their son to be treated has doctors who specialize in Charlie's condition.
Hospital requests new hearing for sick baby Charlie Gard
"We believe, in common with Charlie's parents, it is right to explore this evidence", Great Ormond Street Hospital said in a statement Friday. "Independent medical experts agreed with our clinical team that this treatment would be unjustified", the hospital said.
Calling for her son to be be given the medication, Ms Yates told reporters: "He's our son, he's our flesh and blood".
A Vatican-run hospital in Rome then offered to treat Gard.
Ms Yates added: "If he's still fighting, we're still fighting". Charlie will die from his illness, his doctors have said.
The British parents of a terminally ill baby, facing another court hearing on his condition and care, said Sunday they are hopeful he will receive the experimental treatment that previous rulings have prevented.
The hospital said on Friday it would let the courts re-examine claims that he could be treated, citing "fresh evidence" after US President Donald Trump and Pope Francis drew global attention to the case.
But, after a worldwide media storm over the case, global experts presented Charlie's family with new evidence that gave the baby a much higher chance of survival than previously thought.
"That very institution hired to care for Charlie is trying to strip him of his rights and his parents of their rights to even just take him to get a second opinion".