Indian Father Boby Cheriyil began to face allegations after his elder brother's wife, 43-year-old Shiny Kuriakose, along with her daughters Aleena (11) and Ivana (10), committed suicide by jumping before a speeding passenger train in Kottayam district of Kerala state on February 28.
"The sudden and shocking demise of the family has raised numerous questions and skepticism about the involvement" of the priest now working in Australia's Broken Bay Diocese, said the online petition, launched by a group of Indian Catholics in Australia.
The petition seeking signatures of support asked Church officials and law enforcement agencies to act on their petition as the priest continues in "a position of trust and leadership in the community."
"Numerous news reports and speculations on social media platforms have vociferously raised this issue, demanding an impartial investigation," said the online petition, without specifying any particular allegation against the priest.
More than 17,000 persons have signed the petition as of March 6.
Local news reports said the woman faced domestic violence in her husband's house but did not get support from the priest. Some reports also accused the priest of supporting his brother and foiling the woman's attempt to secure a job, allegedly to stop her from being independent.
The woman is reportedly a qualified medical nurse who left the profession two decades ago. Some reports said hospitals, including some managed by the Church, could not employ her because of her long break from the profession.
Police in Kerala arrested the victim's husband, Noby Luckose, 44, on March 5 on charges of abetting suicide.
Neighbors and the mother of the deceased woman told the media that Cheriyil joined in mentally torturing the woman.
The Kottayam archdiocese, to which the priest and victim woman belong, has so far not responded to the allegations against the priest.
However, an archdiocesan priest, who did not want to be named, said, "The allegations against the Church were unwarranted."
The priest said the woman and her children had been living with her parents for the past nine months, cut off from her husband, his brother-priest, and their family.
"What triggered her to take such a horrible step is something that her family members should answer," the priest told UCA News on March 6.
T. O. Joseph, who leads the Knanaya Catholic Reforms Committee that fights the endogamous practices in the archdiocese, told UCA News the incident proves the Church's lack of interest in the people.
The archdiocese "spend time and energy in defending outdated and useless traditions, but ignores its people. Had the archdiocese cared for its people, the mother, and her daughters would have been alive now," he said.
The hospitals where the women sought jobs included ones managed by the archdiocese. Joseph added that its officials could have helped her with a job if they wanted.
The Kerala State Human Rights Commission has directed the Kottayam district police chief to investigate the circumstances surrounding their deaths and submit a detailed report.
Christians make up 18 percent of Kerala's 33 million people, while Hindus are close to 54 percent and Muslims 26 percent.