Archaeologists who discovered the skeleton beneath a Leicester car park last year decided with backing from the Ministry of Justice that it should be reinterred in the city's cathedral.
But their plan met with a strong wave of opposition from groups based in Yorkshire who claimed that the Plantagenet King should be buried at York Minster instead.
In a hearing on Thursday, Mr Justice Haddon-Cave gave the group, known as the Plantagenet Alliance, permission to launch a High Court challenge in what he described as an "unprecedented" case.
But he added that he hoped the matter could be settled without the need for an "unseemly, undignified and unedifying" legal squabble, urging both sides to "avoid embarking on the (legal) Wars of the Roses Part Two".
He recommended that both parties instead agree to leave the decision in the hands of an independent advisory panel made up of "suitable experts and Privy Councillors" who could make a suitable decision based on submissions from each side.
King Richard III was killed at the Battle of Bosworth in 1485 and was originally buried at Greyfriars church in Leicester, after his body was carried there by supporters of the victorious Henry VII.
The remains were rediscovered beneath a council car park last year by archaeologists from the University of Leicester, and the University was given permission to decide where they should be reinterred.
But while the discovery of the lost king was welcomed by his supporters in York, the University's subsequent decision to rebury the remains in Leicester Cathedral caused outrage and sparked a legal challenge.
Mr Justice Haddon-Cave on Thursday gave the Plantagenet Alliance permission to bring judicial review proceedings against the Justice Secretary and the University of Leicester, with a full hearing to resolve the case expected later in the year.
But in his written judgment he commented: "It is ironic that the Wars of the Roses appear to be returning whence they started – the Temple. Legend has it that John Beaufort and Richard Plantagenet picked the symbolic red and white roses in Inner and Middle Temple gardens.
"I would, however, urge the parties to avoid embarking on the (legal) Wars of the Roses Part 2. In my view, it would be unseemly, undignified and unedifying to have a legal tussle over these royal remains."
He reminded both sides that the archaeological discovery "engages interests beyond those of the immediate parties, and touches on sovereign, state and church", and recommended the formation of an independent panel "who can consult and receive representations from all interested parties and make suitable recommendations with reasonable speed."
Explaining his decision, Mr Justice Haddon-Cave said the context of the King's death and depth of public feeling raised an "obvious duty" to consult widely over how he should be reburied.
Some 26,553 people have signed a petition that the remains should be reinterred at York Minster and 8,115 people have signed a petition that they should be reinterred at Leicester. There was also a "passionate debate" on the matter in the House of Commons in March, he said.
"[The case] involves the remarkable, and unprecedented, discovery of remains of a King of England of considerable historical significance, who died fighting a battle which brought to an end a civil war which divided this country," he said.
"The obvious duty to consult widely arises from this singular fact alone. It was obvious that there would be intense, widespread and legitimate public interest and concern in many quarters as to the treatment and final resting place of Richard III's remains."
The University of Leicester said it was "digesting" the judgment but added that it "continues to take the view that the claim is without merit" and maintains it is "entirely proper and fitting" the remains should be buried in Leicester Cathedral.
"We have received messages of support from some including Michael Ibsen, whose DNA, together with that of another direct descendant of Richard's sister, was key to identifying the remains found at Greyfriars in Leicester," a spokesman said.
Stephen Nicolay, chairman of the Plantagenet Alliance and a 16th great-nephew of the King, said: "We are delighted to receive such a full and comprehensive backing for the case."
He added that the group would be happy to leave the decision in the hands of an independent panel.
"We are not interested in a legal fight, we are just trying to make sure the right thing is done," he said. "We would very much like to discuss this in a sensible and mature manner."
Vanessa Roe, deputy chair of the Plantagenet Alliance, added: "The whole point of bringing it to court was that it was the last thing we could do – they weren't going to discuss it full stop.
"As long as the evidence is looked at objectively, I don't see why an independent panel could not take the decision."