In the health IoT (Internet of Things), the specialized sensor devices can be used to monitor remote health and notify the emergency information, e.g., blood pressure, heart rate, etc. These data can help the doctors to rescue the patients. In cloud-based health IoT, patients' medical/health data is managed by the cloud service providers. Secure storage and privacy preservation are indispensable for the outsourced medical/health data in cloud computing. In this paper, we study the integrity checking and sharing of outsourced private medical/health records for critical patients in public clouds (ICS). The patient can check his own medical/health data integrity and retrieve them. When a patient is in coma, some authorized entities and hospital can cooperate to share the patient's necessary medical/health data in order to rescue the patient. The paper studies the system model, security model and concrete scheme for ICS in public clouds. Based on the bilinear pairing technique, we design an efficient ICS protocol. Through security analysis and performance analysis, the proposed protocol is provably secure and efficient.