Making a distinction between Hinduism and Hindutva, party spokesperson Anand Sharma said that equating the two is "belittling" the philosophy and richness of Hinduism.
"I thought we had one Hitler in making but it seems now we have Two! God save India!," party General Secretary Digvijay Singh, a known detractor of Prime Minister Narendra Modi, tweeted.
A day after the RSS chief made the controversial remarks in Mumbai, Singh asked, "A question to Mohan Bhagwat - Is Hindutva a religious identity? What is its relationship with Sanatan Dharma?
"Does word Hindu or Hindutava figure in any of our Vedas, Upanishads, Gita, Puran or any other of our religious scriptures?," Singh asked.
"RSS should stop fooling the innocent people by using religion in politics. We are proud of our Sanatan Dharm and its tolerance towards others," he said.
Criticising Bhagwat's remarks, Sharma said, "There have been ominous signs of communal terror, violence and polarisation."
He said, "This also confirms that BJP and RSS speak with forked tongue and lack sincerity and conviction towards what Prime Minister Narendra Modi said from ramparts of Red Fort."
"RSS does not have proprietary right over the Hindus or their faith which is tolerant and all embracing," Sharma said.
Congress leader Manish Tewari alleged that there were "concerted efforts" to polarise the country on religious lines.