Counsel for the Milltown Cemetery bomber told the High Court today that the evidence should remain private.
Stone had been imprisoned for waging a sectarian murder campaign expected to keep him behind bars until 2024.
But last month the 65-year-old ex-UDA man was freed on parole from HMP Maghaberry.
The sister of one of his victims is now seeking an urgent judicial review into the lawfulness of that decision.
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Deborah McGuinness’ brother, Thomas McErlean, was among three mourners Stone murdered in a grenade attack on an IRA funeral at Milltown graveyard, west Belfast in March 1988.
Ms McGuinness’ lawyers claim a failure to provide information about parole hearings and to allow bereaved relatives to take part in those proceedings.
As part of the case Northern Ireland’s Parole Commissioners were ordered to disclose reasons for the decision to release Stone.
Mr Justice Colton was told today that a heavily redacted summary has now been provided.
However, Ronan Lavery QC, for Ms McGuinness, said that a dossier of material still cannot be examined.
Citing the principle of open justice, he confirmed: “We want to see exactly what is the medical evidence which seems such a crucial part of this case.”
According to previous reports the former loyalist paramilitary now suffers from a myriad of health conditions.
Counsel for Stone indicated his opposition to further disclosure on human right grounds.
Hugh Southey QC said: “From my client’s point of view the medical evidence is of some importance, and he’s entitled to privacy in relation to that.”
Adjourning the case for four weeks, the judge said a hearing date will be listed at that stage.
Stone, who was also the gunman in three other killings, had previously been freed on licence in 2000.
Six years later he was returned to jail after launching an infamous attempt to murder Sinn Fein leaders Gerry Adams and Martin McGuinness at Stormont.
He entered Parliament Buildings in 2006 armed with explosives, knives and an axe.
Stone denied it had been an attempt to murder the politicians, instead claiming it was an act of performance art.
In 2013 he was told that he must serve the rest of the 30-year tariff on his life sentence.
A series of legal cases have centred on the eligibility of the Parole Commissioners to assess his suitability for release.
In November last year the Court of Appeal overturned a decision that he must remain behind bars until 2024.
Judges held that the six years he spent out on licence should count towards the 30-year tariff, leading to the decision to grant him parole.