Large vision-language models have driven remarkable progress in open-vocabulary robot policies, e.g., generalist robot manipulation policies, that enable robots to complete complex tasks specified in natural language. Despite these successes, open-vocabulary autonomous drone navigation remains an unsolved challenge due to the scarcity of large-scale demonstrations, real-time control demands of drones for stabilization, and lack of reliable external pose estimation modules. In this work, we present SINGER for language-guided autonomous drone navigation in the open world using only onboard sensing and compute. To train robust, open-vocabulary navigation policies, SINGER leverages three central components: (i) a photorealistic language-embedded flight simulator with minimal sim-to-real gap using Gaussian Splatting for efficient data generation, (ii) an RRT-inspired multi-trajectory generation expert for collision-free navigation demonstrations, and these are used to train (iii) a lightweight end-to-end visuomotor policy for real-time closed-loop control. Through extensive hardware flight experiments, we demonstrate superior zero-shot sim-to-real transfer of our policy to unseen environments and unseen language-conditioned goal objects. When trained on ~700k-1M observation action pairs of language conditioned visuomotor data and deployed on hardware, SINGER outperforms a velocity-controlled semantic guidance baseline by reaching the query 23.33% more on average, and maintains the query in the field of view 16.67% more on average, with 10% fewer collisions.