A video circulating online in Muslim-majority Indonesia shows the dog running around a mosque in the West Java district of Bogor while the woman argues with shocked worshippers. Many Muslims consider dogs to be impure.
In the video, the visibly distressed woman says she is Catholic and claims that her husband will be married in the mosque later that day.
She demands an explanation from people in the mosque who apparently have no idea about the purported wedding plans. Doctors have said she should receive psychiatric treatment.
The dog reportedly later died after being hit by a car.
Blasphemy is a criminal offence in Indonesia with a maximum penalty of five years in prison. Rights groups have long called for the abolition of the law, which is often used to persecute Christians and other religious minorities.
Bogor police chief Andi Mochammad Dicky Pastika said the woman is a blasphemy suspect and an investigation is continuing. He said evidence includes a video recording, evidence from five witnesses and her shoes, which she wore inside the mosque.
"We will bring this case to the court," Mr Pastika said. "Even if later the results of the psychiatric examination say that she has psychiatric disorders, as referred to in Article 44 of the Criminal Code, let the judge decide it in court."
The woman was detained at a police hospital, where doctors recommended her transfer to a psychiatric facility.
The woman's identity has not been disclosed. The hospital said sending her to a specialised facility would let her be closer to her family and make it easier to communicate with her.
Amnesty International said the blasphemy case is "unfortunate and absurd" and highlights why Indonesia's blasphemy law should be repealed.
"The state's priority should be her wellbeing. Her actions may have felt insensitive, but these issues can be resolved peacefully, it is not a matter for the courts," said Amnesty's executive director in Indonesia, Usman Hamid.
Religious leaders called on the public to not be incited by the incident.
Yunahar Ilyas, deputy chairman of the Indonesian Ulema Council, the country's top Muslim clerical body, said he doubted the woman understood what she was doing was a threat to religious harmony.
"That makes absolutely no sense if it is done by a sane person in a Muslim-majority nation," he told reporters. "Don't be provoked."
The dog, which was chased out of the mosque, died when it was hit by a car as members of an animal welfare group tried to capture it, one of the group's members said in a Facebook post.
Last year a court in Sumatra sentenced a woman who complained about the volume of a mosque's loudspeakers to 18 months in prison for blasphemy. She was released on parole in May.
Mobs burned and ransacked at least 14 Buddhist temples in Tanjung Balai, a port town on Sumatra, in a riot in July 2016 following reports of the woman's comments.
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