Reconfigurable intelligent surface (RIS) is an emerging technology for future wireless communication systems. In this work, we consider downlink spatial multiplexing enabled by the RIS for weighted sum-rate (WSR) maximization. In the literature, most solutions use alternating gradient-based optimization, which has moderate performance, high complexity, and limited scalability. We propose to apply a fully convolutional network (FCN) to solve this problem, which was originally designed for semantic segmentation of images. The rectangular shape of the RIS and the spatial correlation of channels with adjacent RIS antennas due to the short distance between them encourage us to apply it for the RIS configuration. We design a set of channel features that includes both cascaded channels via the RIS and the direct channel. In the base station (BS), the differentiable minimum mean squared error (MMSE) precoder is used for pretraining and the weighted minimum mean squared error (WMMSE) precoder is then applied for fine-tuning, which is nondifferentiable, more complex, but achieves a better performance. Evaluation results show that the proposed solution has higher performance and allows for a faster evaluation than the baselines. Hence it scales better to a large number of antennas, advancing the RIS one step closer to practical deployment.