Before doing anything rash, have you removed the hub on the other side? inspecting the inner bearing race on that may give you a clue as to the condition of the spindle. The spindle has a lot of mass compared to the bearing race, so it may have survived ok.

Probably the lubricant has just burned and appeared to fuse the parts together. If you see the discoloration in the spindle that would be a sign of over heating, chances are a new spindle AND hub will be needed. A failed inner bearing is usually an indication that the inner seal has failed, and that happens easily if care is not taken when doing anything that required removing the hub.

Steve