Solution: Notch himself has said in an interview that yes, the Redstone blocks in Minecraft allow construction of Turing-complete Machines.

Is an abacus Turing complete?

Yes, an abacus is "Turing Complete" if it is infinitely large and you, as a human being, do the logic in your head. Pen and paper is Turing Complete by the same reasoning (this is ridiculous).

Are Minecraft command blocks Turing complete?

Vanilla Minecraft is most likely Turing Complete due to the combination of command block cloning (for unbounded memory), teleportation (for chunk loading), and block update detection (a component for self-identifying cloning devices).