How can a capacitive touch screen be triggered without human contact?

I haven't actually done this, but it seems the problem is the objects you are using are too small and don't have enough ambient capacitance. A human touching something adds capacitive coupling to the environment. Think of the size and surface area difference between a carrot and a carrot+human.

You should be able to use something conductive that is covered by a thin insulating layer, then connect the conducive part to a conductive plate under the iPhone or to ground. In this case "conductive" only needs to be not a good insulator. As you found, even something like a carrot is conductive enough. Try connecting a ground clip to the other end of the carrot, or connect it to the chassis of your machine.

Basically, a capacitive touch sensor works by detecting the change in capacitance in a "capacitor" that consists of a sensor plate and "ground", where this ground could be your ground plane, or a large nearby conductive plate, or water, or something. The mechanism for capacitance being the ability to hold an electric field means that anything that changes this affinity for electric fields changes the capacitance.

Moisture in the carrot acts as a wire to the human. Conductive material in the Pogo stylus serves the same purpose.

You might want to read Atmel's QTAN0079: Buttons, Sliders and Wheels Touch Sensor Design Guide, which covers designing various types of touch sensors. It goes over the underlying technology. Atmel has several other touch sensor app notes.

Any grounded conductive object should work. We use metal slugs with robots all the time for testing.