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Conventional assembly methods using robots need to change end-effectors or operate two robot arms for assembly. In this study, we propose a screwdriving gripper that can perform the tasks required for the assembly using a single robot arm. The proposed screwdriving gripper mimics a human-two-handed operation and has three features: (1) it performs pick-and-place, peg-in-hole, and screwdriving tasks required for assembly with a single gripper; (2) it uses a flexible link that complies with the contact force in the environment; and (3) it employs the same joints as the pronation and supination of the wrist, which help the manipulator to create a path. We propose a new gripper with 3 fingers and 12 degrees of freedom to implement these features; this gripper is composed of grasping and screwdriving parts. The grasping part has two fingers with a roll-yaw-pitch-pitch joint configuration. Its pitch joint implements wrist pronation and supination. The screwdriving part includes one finger with a roll-pitch-pitch joint configuration and a flexible link that can comply with the environment; this facilitates compliance based on the direction of the external force. The end of the screwdriving finger has a motor with a hex key attached, and an insert tip is attached to the back of the motor. A prototype of the proposed screwdriving gripper is manufactured, and a strategy for assembly using a prototype is proposed. The features of the proposed gripper are verified through screwdriving task experiments using a cooperative robotic arm. The experiments showed that the screwdriving gripper can perform tasks required for the assembly such as pick and place, peg-in-hole, and screwdriving.