Congress president Rahul Gandhi charged Bhagwat with "disrespecting" the Army and those who laid down their lives for the country, saying his remarks insulted every Indian.
"The RSS chief's speech is an insult to every Indian, because it disrespects those who have died for our nation. It is an insult to our flag because it insults every soldier who ever saluted it. Shame on you Mr Bhagwat, for disrespecting our martyrs and our Army. #ApologiseRSS," he tweeted.
He cited the examples of Afghanistan, Iraq, Syria, Somalia and the Congo, and said India could not allow such a development.
"The prime minister must clarify his stand on private militias," he said.
More From This Section
Sharma added that the RSS chief's statement insulted the Indian armed forces "by claiming that in the eventuality of a battle to be fought, the RSS can mobilise itself in three days while the Indian Army will take six to seven months".
"The RSS chief should apologise to the country and the armed forces as he has put a question mark on the capability and bravery of the Indian Army," Sharma said.
The Congress's onslaught came despite the RSS holding that Bhagwat's remarks had been misinterpreted.
The RSS quoted him as saying, "If the situation arises and the Constitution permits, Indian Army would take six months to prepare society whereas Sangh swayamsevaks can be trained in three days, as swayamsevaks practise discipline regularly".
Sharma urged Modi to clarify his stand on the remarks.
Sharma added that the Congress and its workers were opposing such beliefs across the country.
"We expect this mistake to be rectified as it is not in national interest and we take this very seriously as the entire country is disturbed," he said.
"We as a democracy cannot allow any organisation and institution to be given any space when it comes to national defence," Sharma said.
"It disrespects the Constitutional framework. It is an insult to the tricolour," he said.
Congress Communications in-charge Randeep Surjewala said the RSS had insulted the valour of the Army and all Indians rejected their "DNA of division".
"Let all Indians remember - they vilified Gandhi, they slandered Nehru, they slammed Patel, they attacked Ambedkar," he tweeted.
The RSS did not allow women into their shakhas, he said.
"They didnt fly the tricolour for 52 years... Now, Sangh insults Armys valour. We Indians reject your DNA of division," he said.
"Bhagwat ji had said that if the situation arises and the Constitution permits, Indian Army would take 6 months to prepare society whereas Sangh swayamsevaks can be trained in 3 days, as swayamsevaks practise discipline regularly," he said.
"This was (in) no way a comparison between the Indian army and the Sangh swayamsevaks but it was a comparison between general society and swayamsevaks," he said in a statement.