Is there any method or formula that can be used to find the operating voltage of a coil using the resistance of the coil? If so, is it also applicable to AC coils or any other technique?
You cannot determine the operating voltage of a coil in this way unfortunately.
The simplest way I can think of to determine the voltage of an unmarked relay would be to connect it to and adjustable power supply (if DC) or variac (if AC). Then simple turn up the voltage until the relay seems to "click" reliably. I would then turn it up a bit more to provide an operating margin.