This paper introduces a novel method for data-driven robust control of nonlinear systems based on the Koopman operator, utilizing Integral Quadratic Constraints (IQCs). The Koopman operator theory facilitates the linear representation of nonlinear system dynamics in a higher-dimensional space. Data-driven Koopman-based representations inherently provide only approximate models due to various factors. To address this, we focus on effectively characterizing the modeling error, which is crucial for ensuring closed-loop guarantees. We identify non-parametric IQC multipliers to characterize the modeling error in a data-driven fashion through the solution of frequency domain (FD) linear matrix inequalities (LMIs), treating it as an additive uncertainty in robust control design. These multipliers provide a convex set representation of stabilizing robust controllers. We then obtain the optimal controller within this set by solving a different set of FD LMIs. Lastly, we propose an iterative algorithm that alternates between IQC multiplier identification and robust controller synthesis, ensuring the monotonic convergence of a robust performance index. Overall, the proposed framework theoretically ensures asymptotic closed-loop guarantees as the number of available data points approaches infinity. Meanwhile, simulation results demonstrate that the robust performance index converges with a finite number of data trajectories in practice.