For an open-deck steel plate girder railway bridge with rail joints, frequent damage to the bridge members and a high level of noise and vibration occur. By installing continuous welded rail (CWR) to the bridge, it is possible to reduce the noise and impact force of the bridge. However, current girder–sleeper fasteners have low lateral resistance in nature and track buckling can occur when CWR is used on such a bridge. Therefore, a new girder-sleeper fastener with proper lateral resistance to prevent CWR track buckling is needed. In this study, the lateral resistance requirements of a girder-sleeper fastener are investigated through a series of finite element (FE) analyses and parametric study. The effect of peak lateral resistance of the fastener, curve radius, girder length, and lateral displacement of girder are examined. From the analysis results, the peak lateral resistance criterion of the girder–sleeper fastener is proposed for the design of a new fastener for CWR tracks on an open-deck steel plate girder bridge.