No.
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The duration of The New Three Stooges is 1800.0 seconds.
There is no evidence of Johnny Depp in a Three Stooges film .
I'm going to presume your question is, "Did anyone who was later a member of the Three Stooges ever appear in a western with John Wayne?"Shemp Howard appeared in Pittsburgh , which starred Wayne but was not a western.Other than that, no actor from the Three Stooges was ever in a film with Wayne.
It wasn't so much the Stooges that were racist as it was the idiot producers and the idiot film audiences of the era that were.
"Have Rocket, Will Travel" was released August 1, 1959 .