Separating thin, flexible layers that must be individually grasped is a common but challenging manipulation primitive for most off-the-shelf grippers. A prominent example arises in clinical settings: the opening of sterile flat pouches for the preparation of the operating room, where the first step is to separate and grasp the flaps. We present a novel gripper design and opening strategy that enables reliable flap separation and robust seal opening. This capability addresses a high-volume repetitive hospital procedure in which nurses manually open up to 240 bags per shift, a physically demanding task linked to musculoskeletal injuries. Our design combines an active dented-roller fingertip with compliant fingers that exploit environmental constraints to robustly grasp thin flexible flaps. Experiments demonstrate that the proposed gripper reliably grasps and separates sealed bag flaps and other thin-layered materials from the hospital, the most sensitive variable affecting performance being the normal force applied. When two copies of the gripper grasp both flaps, the system withstands the forces needed to open the seals robustly. To our knowledge, this is one of the first demonstrations of robotic assistance to automate this repetitive, low-value, but critical hospital task.