Multi-center disease diagnosis aims to build a global model for all involved medical centers. Due to privacy concerns, it is infeasible to collect data from multiple centers for training (i.e., centralized learning). Federated Learning (FL) is a decentralized framework that enables multiple clients (e.g., medical centers) to collaboratively train a global model while retaining patient data locally for privacy. However, in practice, the data across medical centers are not independently and identically distributed (Non-IID), causing two challenging issues: (1) catastrophic forgetting at clients, i.e., the local model at clients will forget the knowledge received from the global model after local training, causing reduced performance; and (2) invalid aggregation at the server, i.e., the global model at the server may not be favorable to some clients after model aggregation, resulting in a slow convergence rate. To mitigate these issues, an innovative Federated learning using Model Projection (FedMoP) is proposed, which guarantees: (1) the loss of local model on global data does not increase after local training without accessing the global data so that the performance will not be degenerated; and (2) the loss of global model on local data does not increase after aggregation without accessing local data so that convergence rate can be improved. Extensive experimental results show that our FedMoP outperforms state-of-the-art FL methods in terms of accuracy, convergence rate and communication cost. In particular, our FedMoP also achieves comparable or even higher accuracy than centralized learning. Thus, our FedMoP can ensure privacy protection while outperforming centralized learning in accuracy and communication cost.