According to The Guardian, Judge Nicholas Francis has said that only "drastic" new evidence will push him to overturn his decision from April, which granted Great Ormond Street Hospital in London permission to end Charlie Gard's life.
"International attention has been focused on this courageous couple, Connie and Chris, fighting for the life of their son, and I will remain in London calling for the rights of parents to make decisions for their children's care, and for hospital officials to open their doors and let Charlie's parents seek groundbreaking new treatment for their son", Foster said in a statement.
Charlie's parents had previously lost several appeals to bring him to the USA for experimental treatment for his mitochondrial depletion syndrome, a rare genetic disease that has left the boy brain damaged.
British and European courts sided with the hospital's original stance that the experimental treatment was "unjustified" and might cause Charlie more suffering with no cure.
Baby Charlie Gard with his parents, Connie Yates and Chris Gard.
Yates and Gard addressed supporters and the press outside the hospital Sunday alongside Rev. Patrick Mahoney, and said they would continue to fight for the life of their child.
The child's father yelled at a barrister representing the hospital during Monday's hearing, saying "When are you going to start telling the truth?"
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"This treatment has prolonged life in some cases..."
There are 18 children currently on this treatment - one of them wasn't able to do anything and now she's riding a bike, she said.
The baby's mother, Connie Yates, added: "It's really hard". "Until you're in this situation, you don't understand the power of hope", she said.
Francis - who also ruled on an earlier chapter in the case - said everyone involved in the case wanted the best for Charlie.
Meanwhile, the Vatican has said the Pope is following the case "with affection and emotion" and "expresses his own closeness to his parents".
Armstrong claimed that Francis' previous decision against the parents' request in April raises questions about his capability to hear the latest case fairly, though the judge insisted he very strongly wants Gard to live.
When asked if it had made all the difference, Yates said that it had "saved his life so far".
"There is nothing to lose, he deserves a chance", she added.