Large Language Models (LLMs) enhance their problem-solving capability by utilizing external tools. However, in open-world scenarios with massive and evolving tool repositories, existing methods relying on static embedding retrieval or parameter memorization of tools struggle to align user intent with tool semantics or generalize to unseen tools, respectively, leading to suboptimal accuracy of open-world tool retrieval and execution. To address these, we present ToolOmni, a unified agentic framework that enables LLMs for open-world tool use by proactive retrieval and grounded execution within a reasoning loop. First, we construct a cold-start multi-turn interaction dataset to instill foundational agentic capabilities via Supervised Fine-Tuning (SFT). Then, we introduce open-world tool learning based on a Decoupled Multi-Objective GRPO algorithm, which simultaneously optimizes LLMs for both tool retrieval accuracy and execution efficacy in online environments. Extensive experiments demonstrate that ToolOmni achieves state-of-the-art performance both in retrieval and execution, surpassing strong baselines by a significant margin of +10.8% in end-to-end execution success rate, while exhibiting exceptional robustness and generalization capabilities.