In verse 52 of "Song of Myself," Whitman first sees the spotted hawk as an "other," standing outside of himself:
The spotted hawk swoops by and accuses me, he complains of my gab and my loitering.
This swooping hawk, racing past him, is a judge of his behavior, complaining that Whitman is chatting aimlessly and being lazy. The focused, disciplined hawk is suggesting, Whitman thinks, that Whitman get his poem done.
Whitman then likens the hawk to himself, stating:
I too am not a bit tamed, I too am untranslatable,
I sound my barbaric yawp over the roofs of the world.
When we put these two versions of the hawk together, we see that the hawk is a symbol of two sides of Whitman's character. The hawk symbolizes the disciplined part of Whitman, the part we might call the superego that wants to be hardworking, responsible, and do the right thing, not loiter. Yet the hawk also symbolizes the untamable part of Whitman—the part that, like a bird, "yawp[s]" while soaring above the world.
The hawk symbolizes that Whitman is both himself, apart from the hawk, and yet like the hawk. It represents the way that Whitman feels at one with the universe, both as himself and at the same time as a part of the rest of creation. Like the high flying hawk, he too soars, through his poetry, with the untamed exuberance of this bird. He is untranslatable, like the hawk, meaning that there will always a part of himself that can't be pinned down by language.