The two leaders along with Vice President Hamid Ansari, Lok Sabha Speaker Sumitra Mahajan, Defence Minister Manohar Parrikar and Finance Minister Arun Jaitley also attended a reception hosted by Navy Chief Admiral R K Dhowan.
Mukherjee, who is also the Supreme Commander of the defence forces, said the Navy has countered effectively growing challenges and maintained constant vigil at sea.
"The current geostrategic environment has enhanced the role and responsibilities of the Indian Navy in protecting the nation's maritime interests," he said in his message.
Mukherjee complimented the Navy for successful induction of indigenously constructed warships and submarines and its integration of Indian-made equipment and systems, saying it "symbolise India's growing maritime capabilities and quest for strategic self-reliance."
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In his message, Modi said, "On #NavyDay, I salute our Navy personnel and the impregnable spirit & utmost devotion with which they protect our Motherland."
In another tweet, he said, "The steadfast dedication & valour of our Navy Personnel is an unparalleled asset for our Nation."
The Indian Navy observes December 4 every year as Navy Day to commemorate the glorious victory of the Indian Navy in the Indo-Pak War of 1971 and to salute the valiant heroes who brought the country this moment of pride.
At the reception hosted by Navy chief, a photo essay book on Navy was also released by Vice President, the first copy of which was presented to the President.