does a serial port have enough power to drive a comparator circuit?
Your circuit as drawn won't work properly, at least not with an RS232 serial port. The RTS and CTS signals are bipolar with respect to ground, and are expected to meet RS232 levels.

If the RTS line goes low, it will go negative with respect to ground and play tug-of-war with the LM339's ESD protection diodes. You'd also need to handle the case where the RTS line stays high (positive) but the comparator needs to provide a negative voltage on CTS, and therefore there's no guaranteed source of negative voltage, meaning you need to have a positive-to-negative voltage converter.