The Patriarchate condemned "this targeting of innocent civilians and of a sacred place", saying two women and one man had died in the attack on the Holy Family Church.
"This horrific war must come to a complete end," it said, adding the victims had turned to the church compound as a safe haven "after their homes, possessions, and dignity had already been stripped away".
The Holy Family Church spoke in a separate statement of "a number of injured, some in critical condition".
In a telegram for the victims, Pope Leo said he was "deeply saddened" and called for "an immediate ceasefire".
The Pope expressed his "profound hope for dialogue, reconciliation and enduring peace in the region", according to the telegram which was signed by the Vatican Secretary of State Cardinal Pietro Parolin and made no mention of Israel.
The Israeli military said on Thursday an initial inquiry suggested fragments from a shell fired during operational activity in the area hit the church mistakenly and the cause of the incident was under review.
It also said the country did not target churches or religious sites and regretted harm to them or civilians.
Israeli Prime Minister Benjamin Netanyahu said "Israel deeply regrets that a stray ammunition hit Gaza's Holy Family Church" and that "every innocent life lost is a tragedy".
"Israel is investigating the incident and remains committed to protecting civilians and holy sites," Netanyahu said in a statement.
The prime minister's statement followed a call from US President Donald Trump who was "not happy" about the church strike.
"It was not a positive reaction. He called Prime Minister Netanyahu this morning to address the strikes on that church in Gaza," a White House spokeswoman told reporters on Thursday.
"And I understand the prime minister agreed to put out a statement. It was a mistake by the Israelis to hit that Catholic Church. That's what the prime minister relayed to the president," the White House spokeswoman added.
The Patriarchate said the parish priest, Father Gabriel Romanelli, was among those injured, and his church had sustained damage.
Father Romanelli, an Argentine, used to update the late Pope Francis regularly about the Israeli-Palestinian conflict via calls and messages.
TV footage showed him receiving treatment at Al-Ahly Hospital in Gaza, with a bandage around his lower right leg.
Italian Prime Minister Giorgia Meloni blamed Israel for the strike on the religious compound.
"The attacks against the civilian population that Israel has been carrying out for months are unacceptable. No military action can justify such an attitude," she said in a statement.Â
Israeli forces killed at least 27 people in attacks in the Gaza Strip on Thursday, including the three killed in the church strike, according to medics and church officials.
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