The repeated attacks on the hospital in Puthikudiyiruppu has led to the deaths of 12 people in total, according to the ICRC.
The military is said to have surrounded LTTE rebels in the Puthikudiyiruppu jungle region.
Also on Tuesday, the Sri Lankan army said that its forces had captured the LTTE's seventh and final airstrip, effectively grounding the rebel's small air force.
Aid agencies said on Sunday that 250,000 civilians were trapped in areas where heavy fighting was raging. The government puts the number of trapped civilians at about 120,000.
It is unclear how many people have managed to flee the area since then.
The Sri Lankan government has said it cannot guarantee the safety of civilians living among LTTE separatists in the north of the country.
Lakshman Hulugalle, a government spokesman, urged civilians to seek shelter in a government "safe zone", but did not say how the civilians would be able to flee if they were being held against their will as human shields, as claimed by the government's military.
The LTTE denies holding civilians against their will, saying that people were refusing to go since they feared that the army would abuse them.
The shelling of the Puthikudiyiruppu hospital has raised fears for civilians.
Sophie Ramanens, an ICRC spokesperson, told Al Jazeera: "Humanitarian law has to be respected ... we call on both parties to respect their obligations to spare medical facilities."
Morven Murchison-Lochrie, an ICRC medical co-ordinator at the hospital, told the AFP news agency that the staff was "under acute stress, surrounded as they are by the sound of the ongoing fighting and the influx of new patients".
"Ambulances are constantly arriving, but people are also being brought in by wagon, pick-up truck, tractor and even motor scooter."
Brigadier Udaya Nanayakkara, a military spokesman, said that the army was not responsible for the attacks.
"Now the LTTE is firing very desperately everywhere artillery ... one of these shells may have fallen into that area," he said.
The army has declared that rescuing civilians trapped by its offensive against the Tigers is one of its top priorities.
|Hundreds of civilians have been killed or wounded in the fighting since last week
Mahinda Rajapaksa, the Sri Lankan president, had given the LTTE 48 hours to free trapped civilians inside the area, which the LTTE ignored.
The government said it would guarantee safety to people moving during that period, but that it could not ensure that those people who remained among the rebels would be free from danger.
Hundreds of civilians, including children, have been killed or wounded in fighting since last week, the Red Cross has said. Colombo disputes the figures.
The rebel Tamil Tigers have been fighting government forces since 1983, alleging marginalisation by Sinhalese-dominated Colombo.
The group is seeking a separate homeland for ethnic Tamils in the north and east of the country.
More than 70,000 people have so far been killed in the civil war.