"Respect for the Muslims who protected churches at Christmas," Cazeneuve tweeted Saturday. "Harmony and fraternity in the republic."
On hearing that a group of Muslims had gone to a church in the town and the to the cathedral to monitor cars passing by in case of attack during midnight mass and "deliver a message of peace", Ménard tweeted, "Since when have arsonists protected against fires? What is the government doing?"
He claimed that the initiative was organised by two activists who "are known for their fundamentalist, anti-Israeli stances", apparently a reference to leaders of two local groups involved in the vigil.
Local Muslim groups declared themselves "disappointed and saddened", by the mayor's reaction.
Anti-racist campaign SOS Racisme said on Saturday evening that it would file a legal complaint against Ménard, accusing him of "stoking the hatred" that has led to incidents like Friday's trashing of a prayer room in the Corsican regional capital, Ajaccio.
Muslims also staged a protective vigil outside a church in the northern French town of Lens on Christmas Day, at a time when the government stepped up security outside places of worship in the aftermath of the 13 November Paris attacks.
During the state of emergency that followed the attacks, the government has closed
three mosques because of alleged "radicalisation" and plans to strip binationals found guilty of terrorist offences of French nationality.
President François Hollande this week called for "solidarity", "fraternity" and "optimism" as a response to the tragedies.
A rise in Islamophobic incidents has been reported since last January's Charlie Hebdo attacks.
Muslim groups have called on mosques to stage open days on the attacks' aniversary, 9-10 January.