Air carrier Captain reported after push back from gate and prior to taxiing clear; another air carrier taxied in a dangerous and unsafe proximity to aircraft; into the same gate.

Narrative

Pushed off Gate XX. There was an aircraft inbound to the same gate and instead of waiting until we cleared (I was going to taxi out on 1 motor to clear area); the inbound aircraft to Gate XX evasively maneuvered to come into the gate before we cleared. Inbound aircraft taxied through and parallel to the vehicle service road. The right wing tip appeared to come close to our aircraft although it might not have been as close as it seemed; but it was definitely uncomfortable for my First Officer (FO) and I. I report this due to the increased ramp hazards and the need to remain ever so vigilant with tugs; vehicles; personnel; and ingestion risk with possible high thrust events. I have no idea if this aircraft was high thrust; but based on the evasive maneuvers and unusual nose gear positioning to align with J-line of the gate; I might suspect that could have been a factor. Overall; it appeared to be a very unsafe and dangerous maneuver when the best solution would have been to wait until we cleared the area. I also suspect the Flight Crew might have thought we were going to start both engines right there; but even if we did so; the best option would have been to wait. The risk of potential danger for all around was very high and uncomfortable.OAK needs a Ramp Control frequency or better organization of inbound and outbound procedures. When one aircraft pushes off the gate; it might be best to wait until that aircraft clears to allow another aircraft on to that gate.