The outbreak of the Covid-19 pandemic has caused a severe overload of intensive care units with a consequent drastic reduction of ordinary hospitalization, diagnostic tests, and interventions that has led to a significant increase in the number of deaths due to diseases other than the new coronavirus. Information and Communication Technologies (ICT) and the forthcoming Sixth-Generation (6G) system may revolutionize the doctor-patient relationship and healthcare modalities owing to the introduction of eHealth (or digital health). In this context, Non-Terrestrial Networks (NTNs) have vital importance in providing global connectivity thus improving the availability of eHealth services, especially in rural and remote areas. In this paper, we propose a Reliable Satellite Routing Algorithm (RSRA) for a 3D NTN based on Low-Earth Orbit (LEO) satellites. RSRA enables health data detected by sensors worn by the patients to be transmitted to health professionals by selecting a chain of reliable LEO satellites in order to protect such sensitive data. The system-level performance of proposed RSRA algorithm is assessed through simulation campaigns and compared against a routing approach that neglects LEO satellite reliability.