In multilingual healthcare environments, automatic disease diagnosis from clinical text remains a challenging task due to the scarcity of annotated medical data in low-resource languages and the linguistic variability across populations. This paper proposes SwasthLLM, a unified, zero-shot, cross-lingual, and multi-task learning framework for medical diagnosis that operates effectively across English, Hindi, and Bengali without requiring language-specific fine-tuning. At its core, SwasthLLM leverages the multilingual XLM-RoBERTa encoder augmented with a language-aware attention mechanism and a disease classification head, enabling the model to extract medically relevant information regardless of the language structure. To align semantic representations across languages, a Siamese contrastive learning module is introduced, ensuring that equivalent medical texts in different languages produce similar embeddings. Further, a translation consistency module and a contrastive projection head reinforce language-invariant representation learning. SwasthLLM is trained using a multi-task learning strategy, jointly optimizing disease classification, translation alignment, and contrastive learning objectives. Additionally, we employ Model-Agnostic Meta-Learning (MAML) to equip the model with rapid adaptation capabilities for unseen languages or tasks with minimal data. Our phased training pipeline emphasizes robust representation alignment before task-specific fine-tuning. Extensive evaluation shows that SwasthLLM achieves high diagnostic performance, with a test accuracy of 97.22% and an F1-score of 97.17% in supervised settings. Crucially, in zero-shot scenarios, it attains 92.78% accuracy on Hindi and 73.33% accuracy on Bengali medical text, demonstrating strong generalization in low-resource contexts.