Choosing appropriate fabrics is critical for meeting functional and quality demands in robotic textile manufacturing, apparel production, and smart retail. We propose MLLM-Fabric, a robotic framework leveraging multimodal large language models (MLLMs) for fabric sorting and selection. Built on a multimodal robotic platform, the system is trained through supervised fine-tuning and explanation-guided distillation to rank fabric properties. We also release a dataset of 220 diverse fabrics, each with RGB images and synchronized visuotactile and pressure data. Experiments show that our Fabric-Llama-90B consistently outperforms pretrained vision-language baselines in both attribute ranking and selection reliability. Code and dataset are publicly available at https://github.com/limanwang/MLLM-Fabric.