Asserting his secularism, Modi asked Abdullah to reflect on how politics was communalised in Jammu and Kashmir by the National Conference, headed by three generations of his family.
"I want to tell Farooq Abdullah (that) secularism is not only in our constitution but also in our veins," Modi said in a video posted on his official website, with an English translation.
The Bharatiya Janata Party (BJP) leader slammed the Abdullahs for the exodus of the Kashmiri Pandits.
"In our culture of thousands of years, the biggest harm on secular fabric done anywhere (in India) is done in Jammu and Kashmir and it is done due to politics of your father, yourself and your son," said Modi.
In the war of word which started Sunday, Abdullah, a former Jammu and Kashmir chief minister, said Kashmir wouldn't be part of an India that is "communal" and that those who "vote for Modi should jump into the sea".
Farooq Abdullah is the patron of the National Conference, and his son Omar Abdullah is the party president and also the chief minister of Jammu and Kashmir, Farooq's father Sheikh Abdullah was the 'prime minister' of the erstwhile Jammu and Kashmir after its accession to India in 1947. He later became the first chief minister of the state.
Modi said: "It is due to your politics and the politics of your son that Kashmir is the only land from where Pandits were removed solely for their religion. The land of Sufism and harmony has been made communal due to your politics."
"India is a nation that thinks about everyone. Using the sword we have never attacked other cultures. Such expansionist mindset is not in our blood. We cannot tolerate such lessons on what secularism is," he added.
Senior BJP leader Arun Jaitley, meanwhile, suggested the senior Abdullah should take a dip in Dal Lake to repent for not being able to bring back Kashmiri Pandits.
"India will vote Narendra Modi as the Prime Minister. Nobody needs to jump into the sea because of that," Jaitley said in a blog.
"But if Farooq Sahib and his party are silent spectators, when the Kashmiri Pandits cannot go back home, he should, as a gesture of repentance, at least take a dip in the Dal Lake," he said.
BJP spokesperson Nirmala Sitharaman asked why the National Conference did not want secularism in the constitution of Jammu and Kashmir.
"Dr Abdullah must answer why in Jammu and Kashmir, the constitutional amendment making India into a secular republic, is not acceptable to him there," she said.
The word "secular" was added to the preamble of the Indian constitution by an amendment in 1976.
"If that is not done, I think he should refrain from talking about communalism. They have not even taken a step to take back the Kashmiri Pandits, who have been cleansed out of the valley, back into the valley," she said.