The BJP also hit out at Mr Gandhi saying he was trying to "milk a sad incident for political mileage".
In his tweets, Mr Gandhi alleged that Tamils were being brutally killed for not bowing to the Sangh's ideology and blamed the "repressive Manuvadi thoughts" of the RSS and the BJP for beating up of Dalits in Gujarat. He also attached with his tweets a news channel's video clip purportedly showing Dalits being thrashed.
"This soul-stirring video is a result of the Manuvadi thought. Before this disease spreads further in our country, we have to stop this. We will all together defeat this repressive thought of the BJP/RSS. It is time for change," he tweeted in Hindi.
In another tweet, he alleged that the RSS was behind the killing of 10 people in anti-copper plant protests in Tamil Nadu on Tuesday.
"Tamils are being brutally killed because they are refusing to bow to the RSS philosophy. The bullets of RSS and Modi can never crush the spirit of the Tamil people. Tamil Brothers and sisters, we are with you! " he tweeted using the hashtag "SterliteProtest".
Hitting back, the BJP mocked Gandhi for his "immaturity and inexperience" with its spokesperson Sambit Patra wondering what future the Congress will have with such a president, who is "unaware" of India's federal structure.
"You can inherit presidency of a party but you cannot inherit wisdom and understanding," he said, taking a dig at Mr Gandhi for drawing the RSS into the row over the police firing in Tamil Nadu.
In India's federal structure, law and order is a state subject and the BJP-run central government has nothing to do with the police firing, Mr Patra said, asking if Mr Gandhi knew that.
Mr Gandhi so often errs by blaming the RSS for things happening across the country and the BJP would like to gift him Class VI civics book to learn about India's federal structure, Constitution and democracy, Mr Patra mocked.
He wondered if the Congress president would also blame the RSS for anti-Sikh riots of 1984 and the Emergency imposed in 1975 by the then Indira Gandhi government.
"He is not able to give a capable leadership to the Congress and believes that he can milk politically any situation to suit his political opportunism by drawing the RSS and the BJP into the battle," he said.
Mr Patra accused Mr Gandhi of hiding his "failures" behind the veil of attacking the RSS.
The Congress had on Tuesday also tweeted that the gunning down of people in Tamil Nadu in police firing was a "brutal example of state sponsored terrorism".
"These citizens were murdered for protesting against injustice," it had said.
Yesterday, the Congress president had also urged students to "rise", alleging their future was at risk as the prime minister wanted to appoint officers chosen by the RSS by "manipulating" the merit list in civil services examinations.
This message came in the wake of a proposal seeking to bring in major changes in the allocation of services to candidates who clear the all-India examination.
Asked to comment on this, BJP president Amit Shah had on Tuesday dismissed Mr Gandhi's claim, saying he cannot help if the Congress chief sees the Hindutva organisation in everything.