The shower head pipe broke off at the fitting in the wall. Is there a way to get that broken piece out without tearing the wall apart?

If you are a plumber and have enough of the right tools... maybe. Why did it break off? That would give an indication as how likely it can be done or not.

Easiest way is with 1/2" internal pipe wrench. About $10.

I've never been successful at removing the thin walled threads of a shower arm using that tool, Harold, but it is a cheap tool and certainly can't hurt to try it!

Instead, I make a couple small cuts in the threads (not too deep) using a mini. Hacksaw blade and then use a sharp tipped AWL and hammer and undermine the threads of the remaining pipe threads and then pull the threaded part out.

Tricky work, but if Dairycon is careful and doesn't damage the threads of the fitting in the wall, the broken off threads can be removed!

Mark