Totally unrelated.
Scala is a statically typed language. Clojure is a dynamically typed language. This difference shapes both of them fundamentally.
Structural types are static types, period. They're just a way to have the compiler prove statically that an object will have a particular structure (I say prove here, but casting can cause bogus proofs as always).
Protocols in Clojure are a way to create dynamic dispatch that is much faster than reflection or looking things up in a map. In a semantic sense they don't really extend the capabilities of Clojure, but operationally they are significantly faster than the mechanisms used before.
Scala traits are a bit closer to protocols, as are Java interfaces, but again there's a static vs dynamic issue. Scala traits must be associated with a class at compile time, similar to Java interfaces. Clojure protocols can be added to a datatype at runtime after the fact even by a third party.
Something like Clojure protocols is possible in Java and Scala through mechanisms like wrapper/proxy patterns or dynamic proxies ( http://download.oracle.com/javase/1.4.2/docs/guide/reflection/proxy.html ). But those will be a heck of a lot clumsier than Clojure protocols and getting object identity right is tricky as well.
def call(c:{ def call():Unit }) = c.call()
– Germander