Fighting between security forces and Tamil Tiger rebels has escalated since January, when the government withdrew from a 2002 truce arranged by peace broker Norway and stepped up an offensive to capture guerrilla territory.
"The limited number of staff available to care for routine cases alongside war casualties has become a critical issue for several hospitals in northern Sri Lanka," the ICRC said in a statement.
The charity is one of the few still operating inside rebel-held territory.
"The lack of regular deliveries of supplies from the country's health ministry has resulted in clinics and hospitals in the (rebel-held) Wanni running out of basic medicines such as paracetamol, antibiotics and vaccines," the ICRC said.
International humanitarian law applicable in armed-conflict situations made it mandatory for people not involved in fighting to receive medical care as quickly as possible, the ICRC said.
Residents in the northern peninsula of Jaffna were also forced to use mercy