Today, research on healthcare logistics is an important challenge in developing and developed countries, especially when a pandemic such as COVID-19 occurs. The responses required during such a pandemic would benefit from an efficiently designed model for robust and sustainable healthcare logistics. In this study, we focus on home healthcare logistics and services for planning the routing and scheduling of caregivers to visit patients’ homes. Due to the need for social distancing during the COVID-19 pandemic, these services are highly applicable for reducing the growth of the epidemic. In addition to this challenge, home healthcare logistics and services must be redesigned to meet the standards of a triple bottom line approach based on sustainable development goals. A triple bottom line approach finds a balance between economic, environmental, and social criteria for making a sustainable decision. Although, recently, the concept of green home healthcare has been studied based on the total cost and green emissions of home healthcare logistics and services, as far as we know, no research has been conducted on the formulation of a triple bottom line approach for home healthcare logistics and services. To achieve social justice for caregivers, the goal of balancing working time is to find a balance between unemployment time and overtime. Another contribution of this research is to develop a scenario-based robust optimization approach to address the uncertainty of home healthcare logistics and services and to assist with making robust decisions for home healthcare planning. Since our multi-objective optimization model for sustainable and robust home healthcare logistics and services is more complex than other studies, the last novel contribution of this research is to establish an efficient heuristic algorithm based on the Lagrangian relaxation theory. An initial solution is found by defining three heuristic algorithms. Our heuristic algorithms use a symmetric pattern for allocating patients to pharmacies and planning the routing of caregivers. Then, a combination of the epsilon constraint method and the Lagrangian relaxation theory is proposed to generate high-quality Pareto-based solutions in a reasonable time period. Finally, an extensive analysis is done to show that our multi-objective optimization model and proposed heuristic algorithm are efficient and practical, as well as some sensitivities are studied to provide some managerial insights for achieving sustainable and robust home healthcare services in practice.