B737-700
Flight Crew Air Transport Pilot (ATP); Flight Crew Instrument; Flight Crew Multiengine
Flight Crew Last 90 Days 356; Flight Crew Type 6000
Confusion; Distraction; Situational Awareness; Human-Machine Interface; Fatigue
Deviation / Discrepancy - Procedural Clearance; Deviation / Discrepancy - Procedural Published Material / Policy; Ground Incursion Runway; Ground Incursion Taxiway
Air Traffic Control Issued New Clearance; Flight Crew Became Reoriented; Flight Crew Requested ATC Assistance / Clarification
Airport; Environment - Non Weather Related; Human Factors; Chart Or Publication
After landing Tower instructed us to turn right and taxi to gate. As I slowed the aircraft and turned to the right on what I thought was a taxiway; I actually turned onto a runway and then left on a taxiway and the First Officer called Ground to continue to taxi to the gate.  We had briefed to turn on two other taxiways depending on the landing to go to our gate. We slowed quicker than expected. As I looked at the chart on the EFB; I thought the taxiway we were turning on was X.  After clearing I realized it was an X for a closed portion. We continued to the gate and had to hold out. I discussed with the First Officer what had happened. It is a hot spot and I should have realized. I was a little distracted from the strong crosswind landing and calling for the flaps and stowing the speed brake as we exited the runway. Neither Tower nor Ground said anything about the mistake. Maybe have Tower say turn on a specific taxiway not just 'turn right'.  Don't mark a closed portion with an X that can be read as a letter not an X for closure. Better briefing from me.
[Second narrative contains further information.]
B737 flight crew reported they taxied on to a runway instead of the assigned taxiway.