If no oxygen is present, will cut avocados stop browning?
Though I'm sure that it's possible for them to brown due to substances other than oxygen, in general, yes, if you remove oxygen from the cut surfaces of an avocado, it will not turn brown, and yes, if you reduce the quantity of oxygen, browning will slow.  The same works for apples, bananas, and pretty much any other fruit that browns upon exposure to air.

It's not particularly practical for intact avocados, but one method for preventing guacamole from browning is to put plastic wrap right against the surface of the dip, preventing air contact.  Additionally, if you look at packaged guacamole, in addition to the browning resistance caused by acidity, the amount of oxygen in any given container is limited to a few small bubbles.  The oxygen in those bubbles is quickly consumed and the remaining gas does not oxidize the guacamole further.

You could conceivably test this with a vacuum chamber and keep normal slices of avocado green for a prolonged time.  You also could use a food vacuum sealer that bags and evacuates food for long term storage, though ripe avocados are delicate enough that they will probably be crushed into paste by the process.