A gripper is the critical component of the robot end effector for the automatic harvesting of fruit, which determines whether the fruit can be harvested intact or undamaged. In this paper, a robotic gripper mechanism based on three-finger and variable-angle design is designed and analyzed for spherical or cylindrical fruit picking. Among the three fingers of the mechanical gripper, two fingers are rotatable through a pair of synchronous gears to ensure enough contact area for the grasping surfaces, which adapt to fruits of different sizes, such as cherry, loquat, zucchini, and so on. Furthermore, the mathematical relationship between gripper driving force and finger gripping force is obtained by the kinematic analysis of the gripper to realize stable grasping, and a grasping index is employed for the structural parameter optimization of our gripper. The grasping motion is analyzed, and the kinematic simulations are carried out, when the driving speeds of the gripper are 5 mm/s, 10 mm/s, and 15 mm/s, respectively. The system transfer function related to driving speed is obtained by curve fitting. Then, the grasping experiments are conducted with various spherical and cylindrical fruit, of which the weights are between 8 and 300 g and the diameters are from 9 to 122 mm. The experimental results demonstrate that our gripper has good kinematic performance and fruit adaptability. At the same time, the grasping is stable and reliable while no obvious damage appears on the fruit surface.