To focus on the Transcendentalism side, we can look at Whitman's habit of "claiming to be one" with the universal, the universe, and with death.
"I am eternal" is a phrase that both defines Whitman's poetry and the Transcendentalist association with the godhead/Brahman spirit.
To answer this, just think of Whitman's "I Hear America Singing." This is a statement of the democratic impulse in America.
The major theme in the poem is that all these regular average people are what make America. It is people like the hatter and the mason whose songs make America, not the professor and the lawyer and the mayor. This is saying (as do many of Whitman's poems) that America is about the common people, and that is a very democratic statement indeed.
I think that Whitman's greatest contribution to both Transcendentalism and American democracy was in his basic assertion of American freedom the lies within individual choice. Whitman's belief of reclaiming American voice and individual identity in a landscape of growing materialism was something that fit perfectly into Transcendentalism. When Emerson, himself, endorsed it, there seemed to be a convergence between what Whitman was arguing and how the Transcendentalist thinkers flocked to it. Whitman's writing embraces the science and commercialism of industrial America while trying to direct these practical energies toward the “higher mind” of literature, culture, and the soul. This makes for a vision that fulfills the conditions of Transcendentalism and emphasizes that American freedom is universal and applicable to all. Individuals can choose to embrace "higher" or more elevated notions of the good. This becomes vitally important to the idea of democratic self- rule, where individual freedom and autonomy are vital for self- preservation.