Is it obligatory to fast three days as expiation for a vow to buy a stolen gold ring, and should expiation be made for this error?

Buying stolen goods is forbidden because it involves cooperation in sin. You must repent and return the ring to its owner immediately if you know her, and demand the price from the thief. If you do not know the owner, then give the ring or its price in charity on her behalf. Your right to claim from the thief remains. You are not required to fast unless you have made a vow to do so.