The battery of my civic was drained overnight from the AC. Once I took out the fuse the battery stays charged with no issues but I do not have A/C in the car.

I am thinking that It could be the relay that stays with the condition to have it on, and that is what was draining the battery.

Anybody has any other suggestion of what could it be?

 

Your suggestion of checking the A/C Compressor Clutch Relay is where I would start. Does the green A/C light in the dash always stay on? Also check your battery and alternator.