Archie was on life support for four months after he was found with a ligature around his neck at home in April.
His mother, Hollie Dance, fought a long battle against the Royal London Hospital where he was treated, arguing his life support should not be switched off.
But each attempt failed and medication was withdrawn at 10am on Saturday. Archie died just a few hours later.
Ms Dance and the rest of the family have issued a statement through the group Christian Concern, calling for change.
"Yesterday we lost our beautiful boy, Archie. He has fought against all the odds since April, and we are so proud of him," it said.
"We are thankful for the huge amount of support we’ve received from so many different people. We are grateful to our legal team and others who have stood with us as we have faced these difficult challenges.
"We want something good to come out of this tragedy and the horrendous experience we have been put through by the system.
"No parent or family must go through this again. We have been forced to fight a relentless legal battle by the Hospital Trust while faced with an unimaginable tragedy.
"We were backed into a corner by the system, stripped of all our rights, and have had to fight for Archie’s real ‘best interests’ and right to live with everything stacked against us.
"This has now happened too often to parents who do not want their critically ill children to have life-support removed. The pressure of the process has been unbelievable.
"There must be an investigation and inquiry through the proper channels on what has happened to Archie, and we will be calling for change."
Archie had been kept alive by a combination of medical interventions, including ventilation and drug treatments.
In June, specialists said it was highly likely that Archie was "brain stem dead", and that tests conducted showed no discernible brain activity, revealing "significant areas of tissue necrosis".
In a statement yesterday, Barts Health NHS Trust said that treatment was withdrawn in line with court rulings about his best interests.
A spokesperson said: "The trust would like to thank the medical, nursing and support staff in the paediatric intensive care department who looked after Archie following his awful accident.
"They provided high quality care with extraordinary compassion over several months in often trying and distressing circumstances.
"This tragic case not only affected the family and his carers but touched the hearts of many across the country."
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