The judge said evidence from family members had permitted the woman’s “voice to be heard” at the hearing.
He said her “incapacitous state” did not mean that her wishes could be “disregarded”.
Staff at Salford Royal NHS Foundation Trust had asked for a ruling on whether the treatment should continue.
Following the publication of the judge’s ruling on Monday, a lawyer representing one of the daughters said there were no winners or losers.
Mathieu Culverhouse, of law firm Irwin Mitchell, said: “The judge has decided that withdrawing the life-sustaining treatment is in my client’s mother’s best interests given her current quality of life and her previously expressed wishes and feelings.”
“While the court has dealt with the case very sensitively, my client and her sister have found it distressing and painful to have to go to court to fight for their mother’s wishes to be respected, and they hope that in future a different way can be found to resolve cases such as this, so that other families do not have to go through the same ordeal.”