Obama told US lawmakers he had received the letter just a few days ago after Kennedy died on August 25, aged 77, following his 15-month battle with brain cancer.
The Democratic lion had voiced confidence that this year health care reform, which he worked for during almost five decades in the US Senate, would be passed, Obama said.
"He repeated the truth that health care is decisive for our future prosperity, but he also reminded me that it concerns more than material things," the US president said, as Kennedy's widow Vicki, and his two sons, Patrick and Teddy Jr, watched the address.
"'What we face,' he wrote, 'is above all a moral issue; at stake are not just the details of policy, but fundamental principles of social justice and the character of our country,'" Obama said.
In the letter written in May to be delivered after his death, Kennedy told Obama he wanted to "write a few final words to you to express my gratitude for your repeated personal kindnesses to me and one last time, to salute your leadership in giving our country back its future and its truth."
He told Obama, in the text released by the White House, that in his final months the prospect that health care reform was close to being realized "sustained me and the work of achieving it summoned my energy and determination."
Movingly he voiced confidence that "while I will not be there when it happens, you will be the president who at long last signs into law the health care reform that is the great unfinished business of our society."
But Kennedy, accustomed to the rough-and-tumble of Washington politics, warned the young president: "There will be struggles, there always have been, and they are already underway again."
Kennedy's tireless campaign to ensure health care coverage for all Americans was forged from his own experiences, when two of his children were struck with cancer, Obama told the late senator's fellow lawmakers.
"He never forgot the sheer terror and helplessness that any parent feels when a child is badly sick; and he was able to imagine what it must be like for those without insurance; what it would be like to have to say to a wife or a child or an aging parent, 'there is something that could make you better, but I just can't afford it.'
"That large-heartedness, that concern and regard for the plight of others, is not a partisan feeling. It is not a Republican or a Democratic feeling. It, too, is part of the American character," Obama said.
Kennedy, who was the last of the legendary brothers including slain president John F. Kennedy and assassinated senator Robert F. Kennedy, threw his weight behind Obama in last year's White House race.
And in his letter he again bestowed the Kennedy family mantle on Obama saying: "I entered public life with a young president who inspired a generation and the world.
"It gives me great hope that as I leave, another young president inspires another generation and once more on America's behalf inspires the entire world."
The Massachusetts Senate Wednesday meanwhile began debating a change to the state's laws which would allow the governor to appoint a temporary successor to Kennedy before a special Senate seat election to be held in January.
Kennedy's death robbed the Democrats of their 60-seat majority in the 100 seat Senate, and came as Obama seeks to push through health care reform.