An electrosurgical medical device and method for creating thermal welds in engaged tissue. In one embodiment, at least one jaw of the instrument defines a tissue engagement plane carrying a conductive-resistive matrix of a conductively-doped non-conductive elastomer. The engagement surface portions thus can be described as a positive temperature coefficient material that has a unique selected decreased electrical conductance at each selected increased temperature thereof over a targeted treatment range. The conductive-resistive matrix can be engineered to bracket a targeted thermal treatment range, for example about 60.degree. C. to 80.degree. C., at which tissue welding can be accomplished. In one mode of operation, the engagement plane will automatically modulate and spatially localize ohmic heating within the engaged tissue from Rf energy application--across micron-scale portions of the engagement surface. In another mode of operation, a conductive-resistive matrix can induce a "wave" of Rf energy density to sweep across the tissue to thereby weld tissue.