A few things worth noting here. The card named for Pithing Needle is named as it's coming into play, during the resolution of the spell, not when the spell is played. Gorilla Shaman, as mentioned, cannot target a card on the stack. Whenever a spell or ability refers to an "artifact" or "creature," or similar, it means a permanent in play. On the stack it will always be an "Artifact spell" or "Creature spell," and in any zone other than the stack or in play, it would be an "Artifact card" or a "Creature card." Compare the wording on Gorilla Shaman with Artifact Blast and Reconstruction.
Gorilla Shaman

|Creature -- Ape Shaman| 1/1. XX1: Destroy target noncreature artifact with converted mana cost X.
Artifact Blast

|Instant| Counter target artifact spell.
Reconstruction

|Sorcery| Return target artifact card from your graveyard to your hand.
My question is whether or not Gorilla Shaman can destroy a Pithing Needle naming Gorilla Shaman while the needle is still on the stack.
For example, I cast Pithing Needle naming Gorilla Shaman. My opponent then in response taps the appropriate mana for the Shamans ability and targets the Needle.
What happens from here?