Alfie's parents fought the decision to the Supreme Court and before the European Court of Human Rights.
"It is one thing to argue any medical treatment is futile, it is quite another thing to say someone should die because their quality of life is futile".
After the Supreme Court's announcement, a legal representative said the family are to make another appeal to courts in Strasbourg to keep their son alive.
In their written decision, they said there was "no reason for further delay".
The parents want to take their son to be treated at a hospital in Rome and hundreds of people protested outside the hospital last week when his father asked staff to release the boy.
Today the Supreme Court upheld the decision of the High Court and the Court of Appeal confirming that "It has been conclusively determined that it is not in Alfie's best interests to continue to receive treatment or to travel overseas for treatment".
Mr Justice Hayden was told that Alfie, who was born on May 9 2016, was in a "semi-vegetative state" and had a degenerative neurological condition doctors had not definitively diagnosed. The ruling means that a previous court's decision that Alfie be removed from his ventilator on Monday, April 23 remains in effect.
The Supreme Court had also dismissed the suggestion that Alfie was either "detained" or "unlawfully detained".
It is a piece of common law which probably dates back to Anglo-Saxon times.
The spokeswoman said Supreme Court President Lady Hale and two other justices would examine the couple's application.
"Having considered submissions from the parties "on paper", in the usual way, the Supreme Court of the United Kingdom has refused permission for the parents to appeal", said a spokesperson for the Supreme Court.
Giving reasons for the decision, the justices said it was a "desperately sad case".
In February, Mr Justice Hayden ruled that doctors at Alder Hey Children's Hospital in Liverpool could stop treating Alfie against the wishes of his parents following hearings in the Family Division of the High Court in London and Liverpool.
Alfie's parents have done everything in their power to do what they think is best for him even though that is contrary to the views of the doctors.
"Those of us who have to deal with this case dispassionately as a point of law can feel for their sadness".
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