"Once more, we call on both parties to meet their obligation under international humanitarian law to spare at all times the wounded and sick people, medical personnel and medical facilities," the ICRC said.
"We are shocked that patients are not afforded the protection they are entitled to," Sophie Romanens, from the ICRC, told Al Jazeera, adding that the agency was able to evacuate some citizens from the area by sea.
"Our concern now is to be able to continue with the medical evacuation ... there are no medical facilities [in the region]," Romanens said.
The army said on Wednesday that they were not responsible for the attack.
"We did not fire at this location on Monday and it is quite possible that the LTTE attacked them," Brigadier Udaya Nanayakkara, a military spokesman, said.
On Tuesday, the Sri Lankan military said that at least 19 civilians were killed by LTTE fighters while attempting to flee the battle zone.
Another 70 people were said to have been wounded in the alleged attack in the country's northeast.
LTTE fighters could not be reached for their version of the reported incident.
Sri Lanka's north and east has been held by the LTTE as a de facto independent territory, but their control of the area has been significantly reduced by a government offensive during recent months.
The military has now pushed the Tamil Tigers back into a small patch of jungle on the east coast.
The army says that the area under the separatists' control has been reduced to less than 100sq km by the latest military offensive.
More than 20,000 civilians have reportedly fled the war zone in the past few days, heading for government-declared "safe zones" amid continued fighting.
The United Nations and aid agencies say 250,000 civilians are trapped in the conflict zone, but the Sri Lankan military says just 120,000 were in the area when the offensive began.
The LTTE has been battling government forces for about three decades seeking to carve out a separate homeland for ethnic Tamils.