He had suffered from a rare genetic condition called mitochondrial depletion syndrome, which causes progressive muscle weakness and brain damage.
Mr Gard said that before his son's death he had promised the child he would "care for and look after" the youngster's mother.
He wrote on Facebook: "Just a couple of hours before Charlie passed away, I laid on his bed, held his hand and had a little father/son chat with him.
"I told him many things. I told him how much he was loved, how much he would be missed and how proud we all were of him, amongst lots of other things.
"I also made a number of promises to him. Ones that I will never break. And one of those promises I made was that no matter what happens, I would always care for and look after his gorgeous mummy for him.
UCI Center on Stress & Health awarded .3 million by National Institutes of Health