Air Carrier Captain reported on final approach at DCA; a near miss with a helicopter; which was lifting off from a nearby hospital. The proximity of the helicopter resulted in a RA and missed approach.

Narrative

On RNP 19 approach; on the approach path just past waypoint FONVI at about 1100 ft. MSL; we got a TCAS TA. A yellow traffic icon showing 500 ft. below us was just ahead and to the left. As our airplane continued descending on the approach; I; the Captain; looked out my window and did not see anything. We immediately got an RA telling us to 'CLIMB'; at which point we were about 950 ft. MSL. It then called out 'CLIMB NOW' as I was turning off the autopilot and auto throttles and pitching up. We followed our proper procedures; and told DCA Tower that we had an RA we were responding to. ATC then asked if we were still going to land. At this point we were getting uncomfortably close to the prohibited area P56A; so I started turning right absent any instructions from ATC. The First Officer told ATC we were not landing; so Tower eventually gave us a heading and an altitude; which we followed once the TCAS RA had cleared. We then rejoined the approach back at the beginning; speaking to Approach Control; and landed on Runway 19 without further incident. Upon review of the approach path and other information; we estimate we came within 300 ft. or less of what turned out to be a helicopter lifting off of the hospital.