Sr. Gilbert Saliba, the hospital's president, said all patients are treated equally, Israeli or not, because in each one, they see the face of Jesus.
Both priests spoke of their material powerlessness against the weapons of war, but of the importance of their spiritual presence for their community.
The situation in hospitals in the Gaza Strip is dire, and Palestinians are saying that medical supplies will soon run out.
Catholic leaders in the Holy Land called for an end to the cycle of violence and criticized Israel's occupation of Palestinian territories and its collective punishment of Palestinians.
"Using the death of the three Israelis to exact collective punishment on the Palestinian people as a whole and on its legitimate desire to be free is a tragic exploitation of tragedy and promotes more violence and hatred," said a statement Tuesday from the Assembly of Catholic Ordinaries of the Holy Land.
Pope Francis renewed his call for an end to war and terror as he finished his three-day journey to the Middle East with a marathon of meetings.
Most Israelis welcome Francis' visit to the Holy Land as a chance to show the world a side of Israel that supports tolerance and dialogue.
The heads of Christian churches in the region plan to launch an international awareness campaign following a series of anti-Christian vandalism.
A Catholic convent near Jerusalem and a largely Maronite village in Galilee were damaged in recent weeks as a two-year wave of vandalism directed at Christians and Muslims in Israel and the West Bank continued.
In late March, anti-Christian and anti-American graffiti was scrawled on the walls of the Deir Rafat convent, also known as Our Lady Queen of Palestine. The tires of cars at the monastery also were slashed.
Repeated references to persecution of Christians, "usually referring only to what Christians suffer at the hands of criminals claiming to be Muslims, plays into the hands of extremists," said Catholic leaders in the Holy Land.
When Pope Francis visits the Holy Land in May, he will follow the pattern he set last year in Brazil by meeting with the leaders of the three nations he will visit as well as with the less fortunate.
But the trip also will commemorate the 50th anniversary of the historic meeting between Pope Paul VI and Patriarch Athenagoras, and it is for that reason the theme of the trip is: "So that they may be one."