Am I using this LM393A comparator incorrectly?
The LM393A, like most comparators, has an open collector output - that is, the output transistor can pull the output to Ground, but the comparator needs some external device to pull the output high.
You require a pull-up resistor of 10K or so connected from the output pin to your positive supply, instead of your resistor to ground, in order to see the output switch.