A system for implanting a repair device onto a native valve of a natural heart to repair the native valve of a patient during a non-open-heart procedure. The system includes a base assembly having a plurality of pivoting links, a pair of paddles attached to the base assembly, a pair of gripping members attached to the base assembly, and at least one coaptation or spacer element disposed between the pair of gripping members. Movement of the pivoting links of the base assembly causes the paddles to move between an open position and a closed position. The paddles and the gripping members are configured to attach to valve leaflets of the patient. The coaptation or spacer element is configured to fill at least a portion of a gap between the valve leaflets of the patient.