A self-tightening safety tubular clamp for suspending a casing string or other tubular member within a well. In one embodiment, a base has an opening for receiving the casing string. A first gripping member and an opposing second gripping member are each adapted to frictionally engage the casing string. First and second pivot arms include parallel linkages for supporting the gripping members. Each parallel linkage is pivotally secured to the base at one pivot pair, and pivotally secured to one of the gripping members at an upper pivot pair. Frictional contact with the gripping members allows the weight of the casing string to move the pivot arms downward. As the pivot arms move downward, the gripping members move downward and inward into engagement with the casing string. The parallel linkages ensure that gripping surfaces of the gripping members remains vertical, in alignment.