The victim identified as Wahab Abdulai reports stated is receiving medical care at the Legon Hospital when the incident took place on Friday, April 23, 2021.
He is reported to have slipped and fallen from the balcony.
The authorities of the University of Ghana have called for calm.
According to the Dean of Students, Prof Godfred Bokpin, investigations were underway to establish what happened.
He said: "What has happened has happened, but the issue is being investigated, the police, the hall authorities, we are all involved trying to get to the bottom of it, but whilst we are at that, we will urge every student to be calm and keep to their core activities..., and they should not feed into the rumour going around but very soon an official communication will be issued by the university."
The national service personnel is expected to be transferred from the Legon Hospital to the Korle-Bu Teaching Hospital for further treatment.