The A/C circuit is fairly complicated on the Civic and includes a bunch of sensors and switches. Since you can ground the relay coil and the A/C works you can rule out a couple of things like the compressor clutch fuse, relay and wiring to the compressor clutch. Obviously the ECM isn't grounding the relay so let's start there and work backwards. I'll assume you don't have a scantool that can read live data or preform bi-directional control.
The ECM receives a message from the Multiplex Unit (under dash fuse box) when it receives a signal from the A/C compressor thermal protection switch. This (multiplex to ecm) is unlikely to have gone bad and would throw communication codes as well as other codes if it was.
So the thermal protection switch is up next. This is a fairly common failure item on Honda A/C compressors. If you follow the wiring from the A/C compressor there should be a plug with three wires. Two red and one blue, One of the red wires is for the compressor clutch the other paired with the blue wire is for the thermal protection switch. The blue wire leads to the multiplex unit. With the connector unplugged check for continuity. If there isn't any either replace the compressor or bypass the switch.
The thermal protection switch receives a signal from the A/C pressure switch on the red wire. If pressure is too low or high the switch interrupts the signal to the thermal protection switch. Without knowing A/C system pressure this is a guess as to whether the switch is good or bad. As an alternative you can bypass the switch temporarily.
The A/C pressure switch receives a signal from the Heater Control Panel (Connector A pin 4 @ heater control panel). And the Heater Control Panel receives a signal from the A/C on/off switch (Connector B pin 15 @ heater control panel, this looks like it should be 12v or 5v one of the two...). And it looks like the only other input into the Heater Control panel is from the Evaporator Temperature Sensor.
Given the age of the car and not knowing whether the compressor has been replaced before, the part most likely to have failed is the thermal protection switch followed by wiring and the heater control panel. Multiplex units can fail but this is rare.