This paper presents gridless sparse processing for direction-of-arrival (DOA) estimation. The method solves a gridless version of sparse covariance-based estimation using alternating projections. Gridless sparse DOA estimation is represented by the reconstruction of Toeplitz-structured low-rank matrices, which our method recovers by alternatively projecting a solution matrix. Compared to the existing gridless sparse methods, our method improves speed and accuracy and considers non-uniformly configured linear arrays. High-resolution and reliable DOA estimation are achieved even with single-snapshot data, coherent sources, and non-uniform arrays. Simulation results demonstrate performance improvements compared to the existing DOA estimators, including gridless sparse methods. The method is illustrated using experimental data from a real ocean experiment.