A320 Captain reported taking evasive action while taxiing to avoid a commercial airliner that ATC failed to provide separation from; resulted in a critical ground conflict event.

Narrative

After pushing back from gate XX in ZZZ; we contracted ZZZ Ground Control and were instructed to 'Taxi XX; X to Runway XXR'. As we were preparing to turn from XX onto X; we noticed an Aircraft Y crossing Runway XXR on Taxiway Y coming right at us and not appearing to slow down. Although I had not been issued a 'Hold Short'; 'Give Way'; or 'Pass Behind' instruction from the Ground Controller; I decided it would be prudent to stop before Taxiway X and allow the aircraft to pass in front of us and continue up Taxiway XY on their way to the gate. The Aircraft Y contacted Ground Control after crossing the active runway (and crossing our path) to obtain taxi instructions. After the aircraft had passed us; we received a very nonchalant instruction from the Ground Controller to give way to the Aircraft Y (which we had already done in an effort to avoid having a winglet collide with our cockpit and spill my First Officer's coffee). This would be the second of TWO near incursions I would experience on the same day within just a few hours at the same exact intersection due to this poor coordination between the Tower and Ground Controller (once while arriving and one departing). The Ground Controller; then issued us instructions in a rushed; unclear voice; to continue our taxi to Runway XXR via Taxiway X.