Direct-to-satellite (DtS) communication has gained importance recently to support globally connected Internet of things (IoT) networks. However, relatively long distances of densely deployed satellite networks around the Earth cause a high path loss. In addition, since high complexity operations such as beamforming, tracking and equalization have to be performed in IoT devices partially, both the hardware complexity and the need for high-capacity batteries of IoT devices increase. The reconfigurable intelligent surfaces (RISs) have the potential to increase the energy-efficiency and to perform complex signal processing over the transmission environment instead of IoT devices. But, RISs need the information of the cascaded channel in order to change the phase of the incident signal. This study evaluates the pilot signal as a graph and incorporates this information into the graph attention networks (GATs) to track the phase relation through pilot signaling. The proposed GAT-based channel estimation method examines the performance of the DtS IoT networks for different RIS configurations to solve the challenging channel estimation problem. It is shown that the proposed GAT both demonstrates a higher performance with increased robustness under changing conditions and has lower computational complexity compared to conventional deep learning methods. Moreover, bit error rate performance is investigated for RIS designs with discrete and non-uniform phase shifts under channel estimation based on the proposed method. One of the findings in this study is that the channel models of the operating environment and the performance of the channel estimation method must be considered during RIS design to exploit performance improvement as far as possible.