PM Modi was accompanied by Sri Lankan President Maithripala Sirisena and Prime Minister Ranil Wickremesinghe.
After dedicating a 150-bed hospital constructed with Indian aid, he said he is the first Indian Prime Minister to visit "this beautiful region".
PM Modi had visited Sri Lanka two years ago and had undertaken a trip to Jaffna in the north.
The plantation Tamils are distinct from the Tamils of north and east in the island nation, which has for over three decades faced a violent struggle in the north and east by Tamils for their rights.
In a 30-minute speech peppered with quotes from the works of Tamil saints and scholars, he also announced India's assistance in the construction of 10,000 additional houses in upcountry areas, where 4,000 have already been built under ownership basis for the beneficiaries.
He also said that India has decided to extend the 1990 emergency ambulance service, currently operating in western and southern provinces, to other provinces.
"You are the children of Tamil mother. You speak one of the oldest classical languages of the world. Along with that you also speak Sinhalese, which is a matter of pride. Language is not just a tool of communication but it also unifies cultures and people," he said.
Recalling the links between Tamils and Sinhalese, he said they are historically intertwined and were used in the courts of Nayak kings of Madurai and Thanjavur.
"We are for strengthening the spirit of unity and harmony and you must be a force in this," he said, while addressing the Indian-origin Tamils, whose ancestors came to Sri Lanka 200 years ago.
"We see you as a seamless part of our continuum. We want to nurture this relationship between Sri Lanka and India and establish ties in which my government's priority is to work for the progress of all Indians and all Sri Lankans," he said.