Vertebral bone fractures can heal without surgery. But the fact that surgery has been advised suggests there are reasons why conservative (nonoperative) treatment isn’t the best choice for him.
If there are bone fragments present, they can break off and get dislodged. If they move into the spinal canal where the spinal cord is located, serious problems can occur. If the fractures don’t heal, spinal deformity can develop. If the spine starts to compress and curve forward, then kyphosis is the result.
Pain and neurologic symptoms are common with untreated posttraumatic kyphosis. These symptoms often get worse over time. Patients lose function and work capacity declines.
The best results are reported in patients who have the surgery done as quickly as possible. Plates and screws or a double-rod system can be used to hold the spine stable until a complete fusion occurs. Without this treatment, nonunion, deformity, and spinal instability are all potential risks.
If the surgeon has explained all this to your father and he still refuses the operation, it may be helpful to get a second opinion. Perhaps if your father hears the same information from two different sources, he may rethink his options.
The best you can do is make sure he has been advised of the possible negative results of a conservative approach. The choice and the consequences are still his to decide.