Please tell me, could a fractured vertebrae heal itself?

Frequently yes: There are many ways to fracture the spine. Most fractures will heal with time. Often bracing is used to help stabilize the bone to assist healing. There are certain types of fractures where surgery is recommended, but this determination should probably be made by a spine specialist such as an orthopedic surgeon or neurosurgeon.

Yes: Yes, a fractured vertebrae can "heal itself". The question is rather is it in good alignment that it we heal without chronic pain resulting from it, and is there anything about the nature of the injury that would put you at risk for neurological sequela. For that we would need to examine you directly and look at the scans of your specific fracture.