B737 pilot reported during gate pushback they almost collided with an inbound aircraft approaching a gate. Pilot applied brakes abruptly to avoid collision.

Narrative

Cleared to push back and start off Gate XX in the alley at ZZZ. Pushed back tail to the Taxiway on the east line so had to make a 180 degree turn to exit. Cleared by ramp after start to make a 180 on to the west line. As I was halfway through the turn Aircraft Y was taxiing in on the west line to park at Gate either XY or XZ. As I was in the turn had to apply the brakes abruptly to stop as we would have collided with the inbound aircraft. Ramp never said there was an inbound aircraft on the west line and said we were clear to make the turn. The Aircraft Y Captain had to have seen us making the turn and knew they were in our blind spot and made no attempt to stop or slow to prevent hitting us.Cause: Some complacency on the part of the ramp controller should have seen and been told about the inbound aircraft. Not sure if that aircraft was told to hold at the top or was on the wrong line. Very surprised that the Aircraft Y Captain made no attempt to stop!Suggestions: Maybe pushback crew could turn us so the Captain can see before making the turn. I know from now on I won't trust the ramp controller and will be looking for a potential mishap.