Probably not. For the most part, hospitals are not responsible for a doctor's medical malpractice if the doctor is not an employee of the hospital (most doctors are independent contractors and not employees). There are several exceptions to this rule.
To learn more about when hospitals might be liable for a doctor's medical malpractice, see Nolo's article Medical Malpractice: When Can Patients Sue a Hospital for Negligence?
Medical Malpractice Claims
Do You Have a Personal Injury Case?
Settling Your Personal Injury Case
Vehicle Accident Cases
Dog Bites and Related Injuries
Asbestos, Chemicals & Toxic Torts
Dangerous Products & Drugs
When Can Patients Sue a Hospital for Negligence?
Medical Malpractice: Misdiagnosis and Delayed Diagnosis
Medical Malpractice Basics
Medical Malpractice: Common Errors by Doctors and Hospitals