"It is a shocking incident... we won't tolerate it", Nitish said.
He directed state home secretary Amir Subhani to rush to Gopalganj and inquire into the incident.
Stating that stern punitive action would be taken against perpetrators of the crime, he said Subhani would submit a report about the circumstances relating to the murder soon.
Meanwhile, Health Minister Ashwini Kumar Choubey said the state government would bring the Doctors Protection Bill in the upcoming monsoon session of the state legislature seeking to protect doctors.
Prisoners of Gopalganj jail had beaten the jail doctor to death when he had gone inside to treat an inmate.
According to official reports, the doctor was called by an inmate Ranjan Yadav who was suffering from chicken pox and as soon as the doctor had reached, a group of inmates undergoing jail sentence, brutally assaulted him.
The injured doctor was first admitted to a government hospital in Gopalganj but was brought to Patna Medical College Hospital as his condition had deterioted, the report said.
He was declared brought dead at PMCH.
"We have asked the Superintendent of Police (Gopalganj) to ensure speedy trial of the case within a stipulated time-frame", ADG (headquarters) Rajbardhan Sharma said.
An FIR has been registered against Ranjan Yadav and six other inmates, he said adding police investigation was on.