The staff stops being a spell after it is cast and becomes an equipment object attached to the mage instead. Remember that spells create objects, but spells are not themselves objects nor are objects spells. They just are both represented by the same card. So, no it does not itself meet the requirements for the Ring of Healing. The Staff's ability, then, does not identify itself as counting as a spell either, so it also does not meet the requirements.
So my, unofficial, answer is that Faith Healer's Staff does not benefit from Ring of Healing.