Iterative feedback tuning (IFT) enables the data-driven tuning of controller parameters without the explicit need for a parametric model. It is known, however, that IFT can lead to nonrobust solutions. The aim of this paper is to develop an IFT approach with robustness constraints. A constrained IFT problem is formulated that is solved by introducing a penalty function. Essentially, the gradient estimates decompose into: 1) the well-known IFT gradients and 2) the gradients with respect to this penalty function. The latter are obtained through a nonparametric model of the controlled system. This guarantees robust stability while only requiring a nonparametric model. The experimental results obtained from the motion control systems of an industrial wafer scanner confirm enhanced performance with guaranteed robustness estimates.