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John Leckey made the request after an inquest jury found that Daniel Hegarty posed no threat when he was shot dead by the soldier in Creggan in 1972.
He said it was the appropriate course of action given the jury's verdict.
The coroner is legally bound to refer cases to the PPS where it appears an offence has been committed.
The solicitor for the Hegarty family, Des Doherty, said it was a highly significant move.
"It's unprecedented in my experience, in recent years, especially in respect of cases that have a political element, involving the police, the army and civilians," he said.
Daniel, who was a labourer, was shot twice in the head by a soldier close to his home in Creggan. His cousin Christopher, 16, was shot in the head by the same soldier but survived.
The shootings happened during Operation Motorman in July 1972.
Earlier this month, the jury rejected claims that warnings had been shouted to the two teenagers before they were shot.
The operation was aimed at reclaiming "no go areas" in the city from the IRA.
The jury also found that none of the soldiers present attempted to "approach the injured youths to either search them or provide medical assistance".
The 2011 inquest was the second inquest into Daniel's death.
The initial inquest was held in 1973 and recorded an open verdict. A second inquest was ordered by the Attorney General in 2009 following an examination by the Historical Enquiries Team.
The report found that the RUC investigation at the time was "hopelessly inadequate and dreadful".
In 2007, the British government apologised to the Hegarty family after describing Daniel as a terrorist.
