If you had no ground strap at all between the engine/transmission and chassis, you almost certainly would not be able to start the car. The Integra comes with 2 ground straps. I would think that one would generally be enough to at least get the starter to function. Your initial success would suggest that to be the case. The subsequent failure could suggest otherwise. When you say you reconnected it the ground, did you attach it to the original anchor points or use a new location? Did you use a star washer or something similar or was the paint already removed from the area to create a good contact?
Reattaching grounds is definitely one of the easier things to do. After that, I would get a voltmeter, connect the negative lead to the negative terminal of the battery and the positive lead to the positive terminal of the starter and set it up where you can see the display from the driver's seat. Make sure that when you turn the key, the starter is seeing 12 volts ... or something. Seeing no blips of voltage means you probably have a problem with your ignition interlocks or a fuse if you're lucky. Seeing only a few volts or maybe a brief spike and then something barely over 0 volts would suggest a bad ground.