The only direct connection I know between Ray Bradbury and Walt Whitman is Bradbury's borrowing of Whitman's title "I Sing the Body Electric". Bradbury's short story of that title, dealing with an android grandmother, is of course titled in reference to Whitman's poem.
Now there is another connection. Among Bradbury's countless awards there is now a Champion of Literacy award, which was presented to him at the 2nd Annual Walt Whitman Birthplace Association's Benefit for Literacy Gala held at Oheka Castle in Huntington, New York. Of course, Bradbury wasn't there in person, but he gave his thanks via a pre-recorded acceptance speech.
The full story is here.