The US-based International Christian Concern has requested Christians to pray for the believers in Aceh province.
"The law is a part of a trend in the region of stricter and stricter application of Islamic law - an effect that always results in increased hostility against Chrsitians and non-Muslims," a spokesperson for the group said.
Human rights groups in the country have also condemned the new law.
"The laws that have been approved in Aceh are cruel and degrading to humanity," National Commission on Human Rights head Ifdhal Kasim was quoted as saying by AFP.
According to Jakarta Post, the Aceh provincial government official has said it would not sign the controversial Islamic by-law, but also acknowledged that the office of the governor is powerless to stop the law being passed.
The province of Aceh, where Islam arrived from Saudi Arabia centuries ago, is semi-autonomous from the central government.
Separatists in Aceh fought the Indonesian government from 1976 until a peace deal was reached in 2005, ending a conflict that had claimed over 15,000 lives. Since then, its application of Islamic law has become progressively stricter.
According to AFP, the controversial legal change was passed just weeks before a new, more moderate provincial assembly is due to take power. The Assembly will be dominated by the Aceh Party of former separatist fighters of the Free Aceh Movement (GAM).
Indonesia is an overwhelmingly Islamic country. According to the 2000 census, Muslims make up 86.1 per cent of the population, while Christians account for 8.6 per cent.
