For tasks requiring robot-environment interaction, compliant motion is important to ensure stable contact and operational safety. The compliant actuators such as a series elastic actuator and a variable stiffness actuator (VSA) are expected to be one of the promising solutions to provide a compliant motion. In this paper, we propose a gripper actuated by the hybrid variable stiffness actuator developed in our previous work to show how the VSA can improve the ability of grasping. The gripper, which is equipped with two symmetric 4-bar linkages, can conduct the position and stiffness control simultaneously, and the grasping force can be calculated by the estimated torque of the HVSA. The HVSA-actuated gripper is able to grasp fragile objects with compliant motion and the relatively heavy object with stiff motion. Furthermore, the features of the variable stiffness provide operational safety by adjusting the stiffness according to the task. In this study, the relationships between the joint torque and the grasping force, and between the joint stiffness and the Cartesian stiffness are analyzed to control the grasping force. From a series of experiments, it is shown that the gripper can grasp fragile objects such as an egg and a wine glass, as well as relatively heavy object without any soft cover and force/torque sensor.