The car accident was the other driver's fault Can I just settle it myself without an attorney?

You absolutely can. There’s nothing wrong with you doing that. However, the other insurance company has their insured, the other driver’s best interest at heart. They don’t have your best interest at heart. It is not likely that you’re going to get everything that you're entitled to or you could get had you retained an attorney to work on your behalf. While you can do it I wouldn’t recommend it and as easy as making a phone call to a personal injury attorney so that you have someone fighting on your side who has the knowledge and the experience to get you what you deserve after a car accident.