Completing a circuit, how is the necessary voltage determined?
The reasons vary greatly, but I think a good way to summarize them is this: a device does more than simply passively draw current. Often times the actuators and transformers in a device rely on electrostatic phenomena, so in order for it to accomplish a task, a varying range of voltages is required for various devices.
For example, the CRT of a television: the deflection plates use an electric field to deflect the electron beam to a given pixel:

The voltage applied to the vertical and horizontal plates determines the angle of deflection--the greater the voltage, the greater the deflection. So a larger screen would require a higher voltage in order for all the pixels to be reached.
This is just one example, but you get the gist: in most electronic devices the actuators are driven by voltage. Even when they're driven by current, such as with speakers, the signal is usually transmitted throughout the circuit as a voltage.
Another thing to consider is the behavior of transistors. Transistors, which drives all the switches, amplifiers, active filters, and logical elements of a circuit, have a threshold voltage. This is usually fairly small, around 0.7 volts or lower, but if a transistor does not receive at least this voltage on either of its channel terminals, the transistor will essentially be in an &quot;off&quot; state, functionally indistinguishable from a transistor with no voltage applied at all.