There appears to be only one audio recording of Walt Whitman reading his own poetry, an excerpt from the short poem America, originally recorded on wax cylinder. Although the legitimacy of the recording is contested, expert consensus currently holds that it is original and is Whitman’s voice.
America
Centre of equal daughters, equal sons, All, all alike endear’d, grown, ungrown, young or old, Strong, ample, fair, enduring, capable, rich, Perennial with the Earth, with Freedom, Law and Love, A grand, sane, towering, seated Mother, Chair’d in the adamant of Time.
An audio introduction to the recording by Charles Osgood from the book Poetry Speaks: Hear Great Poets Read Their Work from Tennyson to Plath, Elise Paschen and Rebekah Presson Mosby