ABS Warning Light After Spinning Tires

While coming home tonight in a snow/ice storm I got stuck in a drift at the end of our driveway. After some shoveling we got out, but there was some spinning tires before we got enough traction to go. A short time later when we opened the car door there was a warning sound and, after starting the car, the brake/abs warning lights came on.

Is this due to the spinning wheels confusing the brake system, or could there be a bigger problem? If it is the former, what do we need to do to get the lights and sound to reset?

 

Regarding Patrick's question, we tried rocking back and forth (forward, then reverse) to try to get traction. During this process I was told by my passenger who had stepped out of the car that the front wheels were spinning but the back wheels were not.

In response to David's answer, if the system is undamaged and resets on its own as implied, how long does this take?

I can try the OBD-II reset method; the only problem at the moment is finding a paper clip to use so want to find out if there's a chance of it resetting on it's own.