Can an abscessed tooth heal on its own?

NO. Abscess tooth can stop the pain on its own with establishing drainage through a fistula tract on the gum near the abscess tooth, or drain through the gum crevice around the affected tooth. However, continuous draining will cause bone loss either at the apex or at the side of tooth bony wall. It may develop a cyst slowly. It is rather a slow process (a year easily) so you can plan for treatment.
No. You will need to have it treated. Call a local dentist for an exam and consultation to see what exactly is wrong with that tooth.