Will a puppy's dislocated leg heal on its own?

No, dislocated limbs need to be "relocated" in their proper socket before healing can take place. This type of injury is extremely painful. Let dislocated, it can cause damage to the tendons and ligaments which hold the joint in normal position. Get the puppy to a Vet immediately! If you dislocated your shoulder, what would you do about that? I'm sure you'd go to a doctor. Why would you even hesitate about doing the same for your puppy? Anyway, having had a dislocated shoulder, I can tell you that it's painful. Give the pup some relief and get him to the vet. Please. Go to your vet!!