Convolutional neural networks (CNNs) have enabled significant performance leaps in medical image classification tasks. However, translating neural network models for clinical applications remains challenging due to data privacy issues. Fully Homomorphic Encryption (FHE) has the potential to address this challenge as it enables the use of CNNs on encrypted images. However, current HE technology poses immense computational and memory overheads, particularly for high-resolution images such as those seen in the clinical context. We present CaRENets: Compact and Resource-Efficient CNNs for high performance and resource-efficient inference on high-resolution encrypted images in practical applications. At the core, CaRENets comprises a new FHE compact packing scheme that is tightly integrated with CNN functions. CaRENets offers dual advantages of memory efficiency (due to compact packing of images and CNN activations) and inference speed (due to the reduction in the number of ciphertexts created and the associated mathematical operations) over standard interleaved packing schemes. We apply CaRENets to perform homomorphic abnormality detection with 80-bit security level in two clinical conditions - Retinopathy of Prematurity (ROP) and Diabetic Retinopathy (DR). The ROP dataset comprises 96 x 96 grayscale images, while the DR dataset comprises 256 x 256 RGB images. We demonstrate over 45x improvement in memory efficiency and 4-5x speedup in inference over the interleaved packing schemes. As our approach enables memory-efficient low-latency HE inference without imposing additional communication burden, it has implications for practical and secure deep learning inference in clinical imaging.