The AC compressor clutch is turned on by a computer. That computer doesn't know if the compressor is seized up or not, so it is going to cycle the clutch on and off like normal. If the compressor is indeed locked up, the clutch and / or belt will slip and squeal loudly.
All AC systems have a low-pressure cut-out switch or sensor to prevent the compressor clutch from being turned on when system pressure is low. That includes when all the refrigerant has leaked out, so a hole from a rock would simply result in the AC compressor not being turned on. That hole and loss of refrigerant is not going to damage anything else, at least not right away.
All AC systems leak a little refrigerant very slowly, and the system pressures will drop as a result. If the compressor was allowed to run when the system is almost empty, the "low side" could get pumped into a vacuum. If there is a significant leak, that vacuum could cause outside air to be sucked in, along with the humidity in that air. Moisture is deadly to the metal parts when mixed with residual refrigerant. It forms an acid that will attack and corrode those metal parts. That's why a standard part of the procedure, when recharging the system, is to pump it into a nearly perfect vacuum for a good half hour before pumping the new refrigerant in. At a vacuum, water boils at 77 degrees and turns to a vapor that is easier to draw out of the system. The purpose of the low-pressure cut-out switch is to prevent the compressor from running when there is a chance air could be drawn in.
Where potential further damage could occur is when the refrigerant has been lost due to a leak, and you wait a long time to get that repaired. That leaves time for air to go in and out through that leak, and leave the humidity behind where it will form the harmful acid.
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Friday, November 1st, 2019 AT 2:08 PM