Walt Whitman does not have a poem called "A Morning Song", but he does have a poem called "As Adam Early in the Morning". A poem that sometimes goes by a "morning song" was written by Eleanor Farjeon. We can briefly look at both poems here.
Each poem depicts a situation in Eden. Both use Eden as a reference point for a broad metaphor implying that today (contemporary life/the current day) remains fresh, new, innocent and equally participant in God's creation as Eden was/is.
The central idea behind each poem relates to the notion that purity and the perfection of creation remain intact, as in Eden, and that this purity ought to be recognized.
Whitman's poem focuses on the human form in this regard, while Farjeon's poem describes a number of classic examples of mytho-poetic ideas - birds, light, and rain.