Melanoma remains one of the most aggressive forms of skin cancer, and its early detection is critical for improving patient survival. Vision Transformers (ViTs) have recently shown strong performance in dermoscopic image analysis; however, their effectiveness often relies on stacking multiple transformer encoder blocks, resulting in large numbers of trainable parameters and increased model complexity. In this study, we propose TRM-ViT, a parameter-efficient recursive Vision Transformer designed for binary melanoma classification. Instead of using multiple independent encoder blocks, TRM-ViT applies a single transformer encoder block recursively with shared weights, enabling effective depth while substantially reducing the number of trainable parameters. Experiments conducted on the HAM10000 dataset demonstrate that TRM-ViT achieves a ROC–AUC of 0.7952, comparable to a standard Vision Transformer (0.7951), while using approximately seven times fewer parameters (2.15 M vs. 14.57 M). Notably, the proposed model maintains high melanoma sensitivity, making it particularly suitable for screening-oriented applications. These results indicate that recursive weight sharing can provide an effective trade-off between diagnostic performance and model compactness, supporting the development of efficient decision-support tools for melanoma screening in resource-constrained environments.