What are the traits in Walt Whitman's "Song of Myself" that indicates it a kind of secular scripture?
It helps to consider some traits of regular scripture. Regular scripture often contains verses that revere God, praise Him, and delight in all of his good works and creations. They give wisdom and advice on how to bring God closer to you, and how to increase your quality of living. They put God as the most glorious and praiseworthy object of song and verse.
Walt Whitman has a unique ability to take the world around him--the secular, earthly world--and make it seem holy and worth praising. He revels in everday life, in nature around him, in all walks of mankind on earth, in sexuality and the glory of the body, in man's identity, ego and abilities, and in enjoying each and every moment for what it is worth. And throughout "Song of Myself," he sings the praises of all of these "worldly" or secular things. His poetry is like scripture, sanctifying everyday living here on earth, praising its beauty and joy. You can find themes of all of these issues in most of the sections of his poem "Song of Myself."
I hope that these thoughts helped a bit; good luck!