Recent vision backbones, such as Transformer families and state-space models like Mamba, have achieved remarkable progress on image recognition. Despite their empirical success, these architectures remain far from the computational principles of the human brain, often demanding enormous amounts of training data while offering limited interpretability. We propose the Vision Hopfield Memory Network (V-HMN), a brain-inspired vision backbone that integrates hierarchical memory mechanisms across layers with iterative refinement updates. Specifically, V-HMN incorporates local Hopfield modules that provide associative memory dynamics at the image patch level, global Hopfield modules that function as episodic memory for contextual modulation, and a predictive-coding-inspired refinement rule for iterative error correction. By organizing these memory-based modules hierarchically, V-HMN captures both local and global dynamics in a unified framework. Memory retrieval exposes the relationship between inputs and stored patterns, providing a prototype-based form of interpretability through explicit memory retrieval, while the reuse of stored patterns improves data efficiency. This brain-inspired design therefore enhances data efficiency and provides a prototype-based form of interpretability compared to existing self-attention- or state-space-based approaches. We conducted extensive experiments on public image classification benchmarks. V-HMN achieves strong performance on small- and medium-scale benchmarks, and remains competitive with widely adopted backbone architectures on ImageNet despite minimal architectural tuning, while offering improved data efficiency and a prototype-based form of interpretability. These findings highlight the potential of V-HMN as a memory-centric alternative to standard vision backbones, thereby bridging brain-inspired computation with modern machine learning.