We propose the convergent graph solver (CGS), a deep learning method that learns iterative mappings to predict the properties of a graph system at its stationary state (fixed point) with guaranteed convergence. The forward propagation of CGS proceeds in three steps: (1) constructing the input-dependent linear contracting iterative maps, (2) computing the fixed points of the iterative maps, and (3) decoding the fixed points to estimate the properties. The contractivity of the constructed linear maps guarantees the existence and uniqueness of the fixed points following the Banach fixed point theorem. To train CGS efficiently, we also derive a tractable analytical expression for its gradient by leveraging the implicit function theorem. We evaluate the performance of CGS by applying it to various network-analytic and graph benchmark problems. The results indicate that CGS has competitive capabilities for predicting the stationary properties of graph systems, irrespective of whether the target systems are linear or non-linear. CGS also shows high performance for graph classification problems where the existence or the meaning of a fixed point is hard to be clearly defined, which highlights the potential of CGS as a general graph neural network architecture.