Would oxygen levels be stable without photosynthesis?
The answer to your question is in the first graph of the Wikipedia article you linked to: Great Oxygenation Event. In that graph, where time is measures in billions of years, you see that O2, as it was produced, was absorbed in oceans, seabed rock, land surfaces, and finally, about .85 billion years ago, O2 sinks filled and the gas accumulated in the atmosphere.[1] The spike you speak of was probably due to 


  the evolution of the large vascular land plants that brought about increased O2 production and increased O2 levels due to the enhanced global burial of microbially resistant, lignin-rich organic matter during the Permo–Carboniferous (Berner 2004).


More simply put, there hadn't been equilibrium established yet between the amount of oxygen being produced, the stabilization of the ocean layers and the 'burial of excess carbon' in the deepest ocean layers. Most of the O2 on earth was produced many millions of years ago; it is a myth that the O2 we breathe today is the same O2 made by, say, ocean phytoplankton yesterday.

Nadene Unger states in the NYT article that


  [I]t is a myth that photosynthesis controls the amount of oxygen in the atmosphere. Even if all photosynthesis on the planet were shut down, the atmosphere’s oxygen content would change by less than 1 percent.


This is because there are tremendous oxygen sinks that store oxygen gas, such as the surface layers of the ocean. As O2 is used, the ocean would give up free O2. Because of these sinks, some estimate that


  if photosynthesis were to cease, atmospheric oxygen would be depleted in 5000 years due to respiration, weathering, and combustion.[2]


while others state that


  If photosynthesis were for some reason to stop, oxidation of the entire organic carbon reservoir would consume less than 1% of O2 presently in the atmosphere and there would be no further O2 loss (since there would be no organic carbon left to be oxidized). Conversely, if respiration and decay were to stop, conversion of all atmospheric CO2 to O2 by photosynthesis would increase O2 levels by only 0.2%.[3] 


It is very hard to get a comprehensive answer of what would happen exactly. Do we assume all vegetation/phytoplankton suddenly disappears, or does it die (thus increasing to the carbon drain on O2)? Do we assume quick quick death of animals based on lack of vegetation/plankton, etc. as food, or would it be slower? As I said, a comprehensive answer is difficult to find. In any case, the atmospheric O2 would not fluctuate wildly in any immediate fashion.

[1] The oxygenation of the atmosphere and oceans Heinrich D Holland, Phil. Trans. R. Soc., 29 June 2006
[2] The Global Oxygen Cycle
[3] Introduction to Atmospheric Chemistry