I found the solution here: ArduinoISP on the Leonardo does not work on windows · Issue #1182 · arduino/Arduino · GitHub
I just added this to <hardware\arduino\avr\programmers.txt>:
promicroisp.name=Pro Micro as ISP
promicroisp.communication=serial
promicroisp.protocol=arduino
promicroisp.speed=19200
promicroisp.program.protocol=arduino
promicroisp.program.speed=19200
promicroisp.program.tool=avrdude
promicroisp.program.extra_params=-P{serial.port} -b{program.speed}