While all minimally invasive procedures involve navigating from a small incision in the skin to the site of the intervention, it has not been previously demonstrated how this can be done autonomously. To show that autonomous navigation is possible, we investigated it in the hardest place to do it - inside the beating heart. We created a robotic catheter that can navigate through the blood-filled heart using wall-following algorithms inspired by positively thigmotactic animals. The catheter employs haptic vision, a hybrid sense using imaging for both touch-based surface identification and force sensing, to accomplish wall following inside the blood-filled heart. Through in vivo animal experiments, we demonstrate that the performance of an autonomously-controlled robotic catheter rivals that of an experienced clinician. Autonomous navigation is a fundamental capability on which more sophisticated levels of autonomy can be built, e.g., to perform a procedure. Similar to the role of automation in fighter aircraft, such capabilities can free the clinician to focus on the most critical aspects of the procedure while providing precise and repeatable tool motions independent of operator experience and fatigue.