The attack in Nice follows the beheading of Samuel Paty, a Paris school teacher, in an Islamist terror attack earlier this month.
Bishop André Marceau of Nice said that all churches in the city had been closed and would remain under police protection until further notice.
"My sadness is infinite as a human being in the face of what other beings, called human, can do," he said in a statement.
"May Christ's spirit of forgiveness prevail in the face of these barbaric acts," he said.
In a statement, Archbishop Michel Aupetit of Paris commented: "We pray for the victims and their families, but we are stunned by this murderous madness in the name of God. God has revealed Himself as a God of love. Murder in His name is the true, the only, blasphemy, an insult to who He is."
"Christ will tell us in the Gospel of All Saints' Day: 'Blessed are you if you are insulted, if you are persecuted for my sake, for your reward is great in heaven.' We know that these are not just words. From the beginning, Christians have been persecuted, and even today it is they who, although they preach and live by the love of God and neighbor together, pay the heaviest price in hatred and barbarism."
Mohammed Moussaoui, president of French Council of Muslim Faith, condemned the attack and asked French Muslims to cancel their festivities for Mawlid, the Oct. 29 celebration of Muhammad's birthday, "as a sign of mourning and solidarity with the victims and their loved ones."