“The tragedy about this case is that the child is dying and there is nothing to stop that, and for a parent that is the worst thing in life,” said Robert D. Truog, a pediatric intensive care doctor at Boston Children’s Hospital and director of the Center for Bioethics at Harvard Medical School. “What the family needs is to come to terms with this and not to be fighting it. The efforts from the pope and Trump are feeding into their denial about what is going on with their child, and that is neither helping them nor the child.”

The couple’s lawyers declined to comment on Tuesday, and it was not clear if Ms. Yates or Mr. Gard, or the hospital treating their son, would take action in response to the pope or to Mr. Trump, who on Monday weighed in via Twitter. What is clear — based on a review of the extensive legal record in the case, including statements from numerous medical experts — is that the couple have been through extraordinary ups and downs, even by the standards of other families with terminally ill children.

“Charlie’s parents have clearly dedicated their lives to him from the moment that he was born,” a High Court judge, Nicholas Francis, wrote in April, expressing compassion even as he ruled against the parents. Their lives, he said, had been turned into “a living hell.”

Charlie Gard, born on Aug. 4, 2016, was a few weeks old when his parents noticed that he had trouble lifting his head. He was not growing and had to be put on a feeding tube. His breathing became shallow. On Oct. 11 he was admitted to Great Ormond Street Hospital in London, where he has lived ever since.

The baby was given a diagnosis of encephalomyopathic mitochondrial DNA depletion syndrome, which is thought to afflict just 16 children around the world. The disease blocks the mitochondria — the power stations in virtually every human cell — from creating energy. Deaf and kept alive by a ventilator, Charlie cannot grasp objects and suffers from persistent seizures.