Achieving superior outcomes through the use of robots in medical applications requires an integrated approach to the design of the robot, tooling and the procedure itself. In this paper, this approach is applied to develop a robotic technique for closing abnormal communication between the atria of the heart. The goal is to achieve the efficacy of surgical closure as performed on a stopped, open heart with the reduced risk and trauma of a beating-heart catheter-based procedure. In the proposed approach, a concentric tube robot is used to percutaneously access the right atrium and deploy a tissue approximation device. The device is constructed using a metal MEMS fabrication process and is designed to both fit the manipulation capabilities of the robot as well as to reproduce the beneficial features of surgical closure by suture. Experimental results demonstrate device efficacy through manual in-vivo deployment and bench-top robotic deployment.