Could I sue my insurance company because of an accident?

That would be a no.

First of all, can you really prove that you were not at fault? Did you give your insurance company enough credible evidence that they should have questioned the police report? Would their investigation probably lead to their not paying the claim. Did they actually raise your rates so that you have damages, or are you just upset that they paid the other driver/car owner? The case probably involves so little money you could collect that at best an attorney might review the facts with you and write a single demand letter, charging you for about an hour's work. But an attorney would not take the case on on the basis of being paid a percentage of any damages you get, because there is not enough money involved.

Did the police report say you were at fault? The insurance companies can rely on the police report if you don't dispute it.