The Church feels the motive behind these incidents was to fan communal passions in the state known for the religious tolerance.
Goa has witnessed a spate of attacks on religious symbols since early this month. At least nine crosses and a temple have been defiled since July 1 in several areas in South Goa district.
On Sunday night, several gravestones in the Christian cemetery in Curchorem town were allegedly vandalised by unidentified persons.
"The Church in Goa is deeply pained with the recent spate of desecrations of religious structures and burial places of the Christian community, along with a stray incident targetting the Hindu community as well in the state," Archbishop of Goa and Daman Filipe Neri Ferrao said in a statement released here.
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He stated these "violent incidents" seem to be designed by vested interests to provoke communal discord and promote religious hatred.
"While we strongly condemn the same, I personally appeal to our brothers and sisters of all faiths to refrain from taking any retaliatory action or even fanning the flame of religious hatred in their hearts," the Archbishop said.
"At the same time, we urge the government authorities to thoroughly investigate these acts of vandalism and to bring the perpetrators to book," he demanded.
Fr Savio Fernandes, Executive Secretary, Council for Social Justice and Peace, a wing of Goa Church, said they have decided to initiate a fact-finding on these incidents.
"National-level organisations would be arriving in Goa on July 13. They will visit all the places on July 14, where holy crosses were desecrated," he told PTI.
"A fact-finding team (comprising members of a national organisation) will prepare a detailed report after visiting the places and speaking to the people. The report would be then made public," the priest said.
The attacks on the religious places have been condemned by all the political parties with the opposition Congress demanding a CBI probe.
The police are yet to achieve any breakthrough in tracing the vandals.
Fr Fernandes refused to name the national organisation stating that "the council is still in talks with them."
Responding to a question on whom he suspects being behind such acts, fr Fernandes said, "We are not very clear who has done it. We don't have any clue but certainly the fact finding committee will be able to get to it".