The parents were expected to ask judges in a court of appeal April 16 to allow their son to continue life-support.
But Lord Justice Davis, Lady Justice King and Lord Justice Moylan ruled against them after an appeal court hearing in London.
"As has been determined with considerable clarity in this case, Alfie's best interests are determinative and the court has decided what treatment he should or should not receive".
Mr Justice Hayden set a date for the 23-month-old, who is being cared for at Alder Hey Children's Hospital in Liverpool, to be allowed to die.
Moylan also clarified that, in his view, Tom and Kate's parental rights did not give them the right to make decisions about Alfie.
Meanwhile, Mr Evans and Ms James have apologised to families and staff at the hospital following reports of intimidation by protesters.
As news from the court filtered through to Alfie's supporters at the hospital, tears mixed with anger and police officers fanned out around the crowd, who began a chant of "Save Alfie Evans!".
The Court of Appeal, the Supreme Court and the European Court of Human Rights upheld that decision.
Archbishop Vincenzo Paglia, president of the Pontifical Academy for Life, said he strongly hoped there could be an opening of dialogue and collaboration between Alfie's parents and hospital officials so that "together they may seek the integral well-being of Alfie and caring for his life will not be reduced to a legal controversy".
Paul Diamond, Tom and Kate's lawyer, argued that the hospital had violated Alfie's rights by unlawfully detaining him.
Diamond said he will file an appeal with the Supreme Court by Tuesday afternoon, The Liverpool Echo reported.
Last week, Mr Justice Hayden endorsed a detailed plan put forward by Alder Hey doctors for withdrawing life-support treatment, after considering a number of issues at a follow-up High Court hearing.
The parents lost their bid to allow Alfie to be taken overseas for treatment. The hospital issued a statement asking the protesters to lower the volume of their demonstrations and to stop honking their auto horns since it was disturbing some of the patients, according to The Guardian.
"It is my honest hope that everything necessary may be done in order to continue compassionately accompanying little Alfie Evans, and that the deep suffering of his parents may be heard".
"The bad reality was that nearly the entirety of Alfie's brain has been eroded, leaving only water and cerebral spinal fluid", Moylan said, reading from Hayden's previous decision, according to The Sun. Alfie will never make any developmental progress.