Accurate channel prediction and effective beamforming are essential for low Earth orbit (LEO) satellite communications to enhance system capacity and enable high-speed connectivity. Most existing channel prediction and predictive beamforming methods are limited by model generalization capabilities and struggle to adapt to time-varying wireless propagation environments. Inspired by the remarkable generalization and reasoning capabilities of large language models (LLMs), this work proposes an LLM-based channel prediction framework, namely CPLLM, to forecast future channel state information (CSI) for LEO satellites based on historical CSI data. In the proposed CPLLM, a dedicated CSI encoder is designed to map raw CSI data into the textual embedding space, effectively bridging the modality gap and enabling the LLM to perform reliable reasoning over CSI data. Additionally, a CSI decoder is introduced to simultaneously predict CSI for multiple future time slots, substantially reducing the computational burden and inference latency associated with the inherent autoregressive decoding process of LLMs. Then, instead of training the LLM from scratch, we adopt a parameter-efficient fine-tuning strategy, i.e., LoRA, for CPLLM, where the pretrained LLM remains frozen and trainable low-rank matrices are injected into each Transformer decoder layer to enable effective fine-tuning. Furthermore, we extend CPLLM to directly generate beamforming strategies for future time slots based on historical CSI data, namely BFLLM. This extended framework retains the same architecture as CPLLM, while introducing a dedicated beamforming decoder to output beamforming strategies. Finally, extensive simulation results validate the effectiveness of the proposed approaches in channel prediction and predictive beamforming for LEO satellite communications.