Unfortunately, the tree is probably doomed. The nutrients for a tree flow in the layers not too far under the outer bark area. The center of a tree (rings) is mostly inert. So if the damage completely circles the tree, there will be no path for the water and nutrients the tree requires.
I wouldn't give up yet. It looks like the damage isn't too deep. Perhaps you'll get lucky and there are still enough layers of live tissue to recover.
PS: Painting a tree wound is one of those topics that the experts seem to disagree upon. Depending on how valuable this tree is to you, I'd suggest hiring an arborist to come look.
http://www.gardeningknowhow.com/orna...g-on-trees.htm