>>ryandr+(OP)
In short: in some, yes. In my country, one's private insurance company may pay damages for injuries caused by medical malpractice. This may be included in the home insurance or some health/injury/accident insurance. Otherwise and in addition, you are covered by the provider's malpractice insurance. Private medical providers must have malpractice insurance. There is also a national scheme, regulated by law, that covers all public providers, which in practice would be all the emergency departments etc.