2003 Chevy Suburban compressor not engaging

The engine control module controls the relay. If there is no power at the compressor connector, you either have a faulty relay, or the ECM is prohibiting it for some reason. If the A/C COMP, and IGN E fuses are good, do you hear the relay click when the compressor is cycled? The dark green wire at the compressor connecto is power from the relay pin 87. With your key on you should have power at both pins 30, and 86, and the ecm grounds the relay at pin 85. It is possible that you have a low pressure condition that is preventing the ECM to issue a ground for the compressor. The refrigerant pressure sensor is a direct input to the ECM, this is the case on both manual and automatic hvac systems.