B737 Captain reported a tire failed during landing roll.

Narrative

The first officer made a normal landing in the touchdown zone and on speed. During the landing rollout the tower asked us to exit at [Taxiway] 12. At the time we were using automatic braking set to 3; I determined we could make the exit safely and comfortably with manual braking slightly more than level 3. I took the controls and engaged manual braking but not near the decel rate of auto brakes Max. I think this is when the tire ruptured as reported by the crew of air carrier Y that was taxiing in at the time. We rolled out to [Taxiway] 13 and were able to stop clear of the runway on the high speed taxiway. After coming to a stop we requested the emergency equipment; contacted the flight attendants; and briefed the passengers. The jumpseater was helpful in this process taking some of the workload. Once the emergency crew showed up they confirmed the damaged tire and missing gear doors. The gear doors were found on the runway by airport operations. We exited the aircraft via a lift to debrief with maintenance. Maintenance replaced the tire so it could be towed to the hangar.