"Today, again, I plead before you with folded hands to please take immediate steps to recruit more pharmacists to provide relief to the patients," Mr Kejriwal wrote to Anil Baijal.
The letter comes a day after Mr Kejriwal made a surprise visit to Guru Teg Bahadur Hospital in Delhi and found long queues at the pharmacy.
"My intention is not to find faults or point fingers," it read.
Last week, vacant posts in the hospitals became a flashpoint between Lieutenant Governor Baijal and the chief minister with both blaming each other for the delay in making appointments.
Mr Kejriwal had told Anil Baijal that 2,169 posts were lying vacant in hospitals as the latter had approved a decision to hire only retired personnel, and added that not enough retired personnel were available.
He alleged that the elected government was being kept in the dark about the decision and said: "I would also be grateful if necessary directions are issued to all officers to stop hiding files from the elected government."
Responding to the Chief Minister, Lieutenant Governor Baijal had said that the proposal to appoint retired employees to the vacancies was moved by the Health and Family Welfare Department on directions of the Chief Minister in a meeting on August 8, 2017.
"The minutes of that meeting were also issued by the CM Office. Therefore, it is surprising that the Chief Minister has now raised questions regarding the filling up of vacant posts with retired personnel," the Lieutenant Governor had said.