Electronic – arduino – How to make a micro-controller act as a finger on a touch screen

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.

so that means that I should be able to interact with it as long as I can bring a small current to the screen where I want to touch.

No. You don't "bring" current to the device. These devices measure capacitance, not current or voltage. The additional capacitance thru your body back to ambient or circuit ground is detected by looking at how the capacitive pad responds to some specifically chosen signals.