Not really. Your yeasts actually depend on the small amount of oxygen in the headspace to multiply and convert the priming sugar into CO2 and a little more alcohol to carbonate up your bottles. This headspace then quickly fills with CO2 gas as a byproduct of priming sugar fermentation. The pressure of the CO2 that builds in the space forces more CO2 into liquid suspension, so having just a small amount of headspace is somewhat desirable. I use longneck bottles and fill mine to the top with a bottling wand. When I pull the wand out of the bottle after it's full, I have about 1" of headspace in the neck which is just about perfect.
The only exception to this is when you force carbonate using compressed CO2. These beers don't depend on any bottle conditioning to carbonate the beer as CO2 is pressurized into them using a CO2 tank and a regulator. This is typically done in a keg first under moderately high CO2 pressure and then beer is filled into individual bottles for sharing/judging/etc. using something along the lines of a beer gun (or bottle filler in a commercial brewery). These beers are best capped "on foam," meaning the bottles should be filled completely to the point of foam dripping down the sides to eliminate any headspace in the bottle.
The reason for this is oxygen exposure on beer that's already carbonated just promotes oxidation, as there isn't any priming sugar for yeast to consume the remaining oxygen. So, they fill the bottles and kegs as full as they can.
I don't mean to criticize, I just would like to comment here. Yeast is known for its ability to undergo anaerobic (without oxygen) respiration. Yeast can go about its yeast business of producing alcohol and CO2 without oxygen.
I've read that this is one of the great things about yeast for brewing and rising dough. Effectively, the organism is more versatile than other forms of life.
I've also read that all yeast activity in brewing should be anaerobic (no definitive source on this). If the yeast is using oxygen to produce energy, then the beer will not be as good. The only oxygen that should be consumed by the yeast is during initial fermentation to provide the organism with the necessary oxygen nutrient to replicate its cell wall and copy itself as it reproduces.
By these findings, bottle head-space should not require any oxygen for the existing yeast to produce alcohol and CO2 without replicating itself.