What is the ruling regarding a doctor who sent a female patient home despite there being indications that necessitated her transfer to the hospital, and the patient died the following day?

If you were negligent in transferring the patient to the hospital when it was necessary, then you have sinned. You must hasten to repent to Allah by feeling remorse and resolving not to return to that sin. Follow a bad deed with a good one, for Allah Almighty is Forgiving to whoever repents, believes, and does righteous deeds. You are not liable for what happened.