Baby Alfie's parents lost legal challenges against a High Court ruling allowing the hospital to withdraw ventilation.
The court ruled there was nothing more doctors could do.
Thomas wrote, "My gladiator lay down his shield and gained his wings at 02:30".
Evans had been in a semi-vegetative state for over a year after being diagnosed with a mysterious neurodegenerative disease associated with severe epilepsy, and had been kept alive in Alder Hey Hospital's critical care unit via ventilator since December 2016.
Before his death a prayer vigil for Alfie Evans was held at the Vatican.
British law states that parents "cannot demand a particular treatment to be continued where the burdens of the treatment clearly outweigh the benefits for the child", Agence France-Presse reported.
The case has stirred up the issue of whether judges, doctors or parents have the right to decide on a child's life.
The pontiff had shown support for the efforts of the parents of the toddler to get him help outside of the UK.
The Bradley Lowery Foundation, named after the six-year-old boy who died after suffering from neuroblastoma, paid tribute to Alfie's parents, saying "they did everything they could for their little boy".
Francis has spoken repeatedly about Alfie.
It was a strikingly different tone from the one he struck earlier, when he said doctors were wrong about Alfie's prognosis and threatened to resume his fight in court.
The Pope, who gave Alfie and his parents Italian citizenship as they tried to win permission to move him to Rome for treatment, said: I am deeply moved by the death of little Alfie.
Both cases were also influenced by opportunistic elements - in the Gard case, USA lawmakers intent on rubbishing Obamacare, and in the Evans case, a controversial Christian pro-life body that is funding the legal challenge.
"I wish to repeat and strongly confirm that the only master of life from the beginning until the natural end is God". "It is our duty to do all that is possible to safeguard life".
Hospital bosses say they are "liaising directly" with the family after disagreements over his treatment.
Doctors overseeing Alfie's care in the city of Liverpool said further treatment was futile and not in his best interests, and that he should be allowed to die.
"I know the date and time my child is going to die and we haven't even had time as a family to come to terms with this".
He thanked the hospital staff "for their dignity and professionalism during what must be an incredibly hard time for them too".