In addition to what was said in answer #1, I would like to point out that something else Emerson wrote in his famous essay "The Poet" had a very powerful influence on Walt Whitman and subsequently on all modern poetry. According to Emerson:
For it is not metres, but a metre-making argument, that makes a poem, — a thought so passionate and alive, that, like the spirit of a plant or an animal, it has an architecture of its own, and adorns nature with a new thing. The thought and the form are equal in the order of time, but in the order of genesis the thought is prior to the form.
Whitman took this advice literally, and much of his poetry in Leaves of Grass shows that he was ignoring many of the traditions of European poetry, most of which had themselves been derived from the ancient Greeks and Romans. Walt Whitman's poetry notably disregarded meter, rhyme, and line length. He also seemed to pay scant attention to such poetic devices as similes and metaphors. His poetry represented a revolutionary departure from English poetry by such great names as Shakespeare and Keats. Here is a sample of Whitman's characteristic style from his "Song of Myself":
My tongue, every atom of my blood, form’d from this soil, this air,
Born here of parents born here from parents the same, and their parents the same,
I, now thirty-seven years old in perfect health begin,
Hoping to cease not till death.
Hardly anything remains of traditional poetry in such lines. Every line is of a different length. There are no rhyming words at the ends of the lines. There are no metaphors or similes. There is no detectable meter. There is something in most of Whitman's poetry that makes it sound, somehow, like poetry. It is a sort of chanting. Note how he repeats the word "parents" three times in the second line of the above sample. Whitman is obviously following Emerson's idea that it is not meter but a meter-making argument that makes a poem. "Form follows function."
Many anthologies of modern poetry begin with selections from Walt Whitman. At first he was scorned by European and even American critics. Henry James published a sarcastic review of "Leaves of Grass" in which he wrote, among other things, that if you are going to write prose you should not break it up into separate lines and should use orthodox punctuation. But modern poetry breaks all the old rules in the belief that the form should follow the thought and not that the thought should have to be forced into a rigid predetermined form.
Emerson stated the idea and Whitman provided the early examples. Emerson was quite pleased with Whitman's poetry and praised him for his efforts. With the great Ralph Waldo Emerson as his advocate, Whitman became famous in his time. It is probably only fair to call Whitman the father of modern poetry.