An inquest was halted in October when coroner Alan Wilson referred the boy's death in 2014 at the vicarage in Freckleton, Lancashire to prosecutors.
A doctor had told the hearing Jonathan could have survived with resuscitation.
The Rev James Percival and his daughter Ruth were initially held on suspicion of murder but no charges were brought.
Ms Percival, 30, gave birth in a bathroom at the vicarage and her 66-year-old father, then vicar of Holy Trinity CE Church, helped her.
He told police the baby appeared "sallow and lifeless" and he believed the infant was "obviously deceased".
However, Blackpool Coroner's Court was told the boy could have survived up to 15 minutes after delivery on 25 November 2014.
Consultant neonatologist Dr Ruth Gottstein told the inquest that when babies are born with the umbilical cord around their neck, there is an 80% survival rate following resuscitation.
As well as being initially arrested on suspicion of his murder, both Mr Percival and his daughter were held on suspicion of conspiring to conceal the birth of a child.
The baby was not seen by a medical professional for about two hours and was left alone in the house wrapped in a towel on a sofa as the pair visited their GP, the inquest was told.
In a statement, the CPS said: "There remains insufficient evidence to provide a realistic prospect of conviction for a criminal prosecution.
"The CPS has written to the coroner to explain the decision and to confirm the inquest can now be continued."
No charges will be brought over the death of a baby at a vicarage because of "insufficient evidence", the Crown Prosecution Service (CPS) has said.