No.

Having the A/C light on does not mean the compressor is activated. It means the compressor may be activated if the climate control determines it should be, e.g., if you have the auto temp set at 78 degrees, the outside air is at 45 degrees, and you do not have the defroster setting turned on, the A/C compressor will not be engaged, even if the green light on the A/C button is lit.

If you have the defrost setting activated, the system will activate the A/C, as it assume you would like to be able to see through the windshield ASAP, and dehumidified air will clear the windscreen faster.