The incident happened at Alder Hey Children's Hospital
The doctor picked up the wrong syringe and re-injected Aaron Savage with a muscle relaxant drug instead of a saline solution.
Aaron's parents refused to let treatment continue at Liverpool's Alder Hey hospital following the incident.
The baby, who has a chronic lung disease, has now been transferred to the Liverpool Women's Hospital.
His father Andy Savage said: "He is a fighter, there is no doubt about that, they're doing all they can to make him comfortable, but under the circumstances each day is a bonus."
Alder Hey hospital has apologised for the mistake and introduced colour-coded syringes as a result of the incident.
Mr Savage said Aaron needed treatment to enlarge his airways to ease his breathing.
He needed to be kept still using muscle relaxant.
'Handling error'
"We wanted him [ Aaron ] removed from Alder Hey... we wanted the doctor to have no dealings with him whatsoever, and we also requested a letter from the hospital stating that the drug had no and would have no ill effects on Aaron," he said.
In a statement the Royal Liverpool Children's NHS trust said: "During Aaron's treatment in the intensive care unit, a handling error by a junior doctor did occur which resulted in a bigger dose of muscle relaxant being administered to Aaron than had been planned.
"However, the trust would like to stress that this did not in any way affect Aaron's condition.
"Aaron's parents have received a full explanation, and the trust has apologised for any additional stress caused.
"There has been a full review of the incident which has resulted in new procedures being put in place in order to ensure that this will not happen again," the statement went on.