Home Minister Rajnath Singh also paid tribute to those who served the nation during Operation Vijay.
President Kovind said every Indian acknowledges the efforts and gallantry of India's armed forces.
"We salute the ultimate sacrifice of the martyrs of Kargil, and record our everlasting debt to their families," he tweeted.
Modi also tweeted: "On Kargil Vijay Diwas, a grateful nation pays homage to all those who served the nation during Operation Vijay. Our brave soldiers ensured that India remains protected and gave a befitting answer to those who tried to vitiate the atmosphere of peace."
Hailing former Prime Minister Atal Bihari Vajpayee, he said, "India will always remember with pride, the outstanding political leadership provided by Atal ji during Operation Vijay. He led from the front, supported our armed forces and clearly articulated India's stand at the world stage."
Home Minister Rajnath Singh added: "On Kargil Vijay Diwas, we salute the unflinching courage and supreme sacrifice of all those soldiers who fought valiantly in 1999. Every Indian citizen is proud of their heroism and service to the nation."
"During the Kargil War in 1999, I had the opportunity to go to Kargil and show solidarity with our brave soldiers," he said on Twitter.
The Prime Minister said that in 1999, he was working for the BJP in Jammu and Kashmir and Himachal Pradesh.
"The visit to Kargil and interactions with soldiers are unforgettable," Modi said.
In the photographs, he is seen interacting with army men and meeting wounded soldiers.
During the Kargil War in 1999, I had the opportunity to go to Kargil and show solidarity with our brave soldiers.
This was the time when I was working for my Party in J&K as well as Himachal Pradesh.
The visit to Kargil and interactions with soldiers are unforgettable. pic.twitter.com/E5QUgHlTDS
— Narendra Modi (@narendramodi) July 26, 2019
"A grateful nation acknowledges the gallantry of our Armed Forces on the heights of Kargil in 1999. We salute the grit and valour of those who defended India, and record our everlasting debt to those who never returned," the President tweeted.
The Prime Minister expressed his heart-felt gratitude. "This day reminds us of the courage, bravery and dedication of our soldiers. On this occasion, my humble tribute to those mighty warriors, who gave up their all to defend their motherland," he tweeted in Hindi.
"Let's pay homage to the Kargil martyrs and salute the courage, valour and sacrifice of our bravehearts. Let's remember the supreme sacrifice made by the brave soldiers of India, to uphold and safeguard the integrity of the nation," said Indian Air Force.
"26 July immortalised as Kargil Vijay Diwas is saga of glorious victory of the Nation during Kargil Conflict in May-July 1999. #IndianArmy soldiers fought legendary battles in Drass, Kaksar, Batalik & Turtok sectors. #Salute to the courage, valour & sacrifice of our martyrs & heroes," tweeted Additional Directorate General of Public Information, IHQ of MoD (Army).