Graph Convolutional Neural Networks (GCNs) have gained widespread popularity in various fields like personal healthcare and financial systems, due to their remarkable performance. Despite the growing demand for cloud-based GCN services, privacy concerns over sensitive graph data remain significant. Homomorphic Encryption (HE) facilitates Privacy-Preserving Machine Learning (PPML) by allowing computations to be performed on encrypted data. However, HE introduces substantial computational overhead, particularly for GCN operations that require rotations and multiplications in matrix products. The sparsity of GCNs offers significant performance potential, but their irregularity introduces additional operations that reduce practical gains. In this paper, we propose FicGCN, a HE-based framework specifically designed to harness the sparse characteristics of GCNs and strike a globally optimal balance between aggregation and combination operations. FicGCN employs a latency-aware packing scheme, a Sparse Intra-Ciphertext Aggregation (SpIntra-CA) method to minimize rotation overhead, and a region-based data reordering driven by local adjacency structure. We evaluated FicGCN on several popular datasets, and the results show that FicGCN achieved the best performance across all tested datasets, with up to a $4.10\times$ improvement over the latest design.