Nupur Sharma, an Indian woman, is accused of making derogatory remarks about Prophet Muhammad.
Delhi police had opened investigations against Ms Sharma and a few others.
Investigations have been launched against those attempting to disrupt public order by instigating people along partisan lines.
BJP suspended Ms Sharma from the party
The head of the party's Delhi media unit was also expelled for sharing a screenshot of her offensive comment in a tweet.
Ms Sharma, a law student at Delhi University, began her political career in 2008 when she was chosen president of the students' union.
Her political career picked up pace in 2011 when she returned to India after doing her masters in international business law.
Her ability to argue and present her point of view in both English and Hindi earned her a spot on the BJP's media committee for the Delhi assembly elections in 2013.
When new elections were called two years later, she ran as the BJP's candidate against Delhi Chief Minister Arvind Kejriwal.