Air carrier captain reported a ground conflict with another air carrier during taxi.

Narrative

After landing in ZZZ we had been instructed to taxi on taxiway 1 and hold short of 2. Our arrival gate was gate A and our planned entry point taxiway 3. Due to heavy traffic we sat for a few minutes at taxiway 2 before being told by ground to 'follow the traffic ahead'. The traffic in front of us was another inbound company aircraft heading to different gates. His taxi path was continuing on taxiway 1 followed by a turn on taxiway 4 followed by taxiway 5 to the ramp. So we continued on taxiway 1 to 4. We contacted ramp and were told by the ramp controller that gate A was open and to enter taxiway 3. Taxiway 3 is directly abeam taxiway 4; so our plan was to continue to follow company traffic onto taxiway 4 and then go straight into the ramp. After getting clearance from ramp I wanted to make sure ground knew we were cleared in on taxiway 3; so I called ground and told them 'Aircraft X is cleared in to taxiway 3'. The ground controller responded with 'Taxi into taxiway 3'.We slowly started making the right hand turn onto taxiway 4 and I looked left and stated 'clear left'. My FO looked out his window and stated 'Wait; we've got traffic on the right that isn't stopping.' I stepped on the brakes and came to a stop about halfway through the turn and saw Aircraft Y taxiing at high speed on taxiway 6. The CRJ did not stop to give way to us; in fact it didn't even look like he slowed down. Fortunately we stopped early enough to allow for him to pass in front of us; once the CRJ was clear we continued ahead into the ramp. No further issues occurred after.At no point did either ground or the CRJ say anything about this possible collision. The only person to speak up and notice a possible collision was my FO. It's thanks to his vigilance and quick thinking that allowed for me to stop early enough to make sure we didn't collide.