What's the deepest reason why QCD bound states have integer charge?

The simplest answer to your question is a quite old idea, captured best I think by the Rishon model of Haim Harari, Michael Shupe, Nathan Seiberg, and others.

Their answer is the simple and rather obvious one: Hadrons and leptons have identical charge because they are composed out of the same set of more fundamental particles and anti-particles, specifically an uncharged V particle and a one-third charged T particle.

Alas, in terms of mathematical development the Rishon model is more akin to an intriguing speculation than a fully developed and predictive physics model. I do not personally think that any particle-based version of the Rishon model can ever be made to work. My suspicion is that theories like the Rishon model are best viewed as incomplete and distorted images of some far less obvious form of composition, one with components that conserve certain properties but cannot be called particles in any traditional meaning of the word.

Nonetheless, the Rishon model strikes me as orders of magnitude better than some of the more recent trends to explain issues such as electron-proton charge equality by invoking what amounts to anthropic self-selection gone wild. Why? Because Rishon theory at least tries to explain astonishing coincidences. If Newton had given up so easily on looking for deeper roots behind an effect as infinitely precise and in-your-face obvious as electrons and protons have identical charge magnitudes, we'd still be talking about how amazing and lovely it is that Great Angels push the planets around in patterns too lofty and subtle for humans ever to understand.

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2012-09-27 Addendum

Here's a point I should make clear for the record, since I came down pretty heavy on the idea that evolving universes could create balanced sets of charges via nothing more than the anthropic principle.

The anthropic observation that the existence of life as we know it seems to require that many fundamental constants to be very tightly constrained and balanced with each other is a simply delightful observation that truly needs explanation. Simple examples include such things as the remarkably long an sharp ridge of stable isotopes that enable complex chemistry, nuclear fusion suitable for stars, and the ability of carbon (with nitrogen and other helpers) to form indefinitely long stable chains. These applications of the anthropic principle are all in effect fine-tuning issues, and I think they are entirely legitimate issues for applying your own personal favorite version of anthropic selection if you are so inclined.

Where I have deep heartburn is with the far more radical versions of the idea that essentially toss all aspects of physics into one big mysterious anthropic pot that then magically burps out whatever it is you need to make life possible. If that is true, why do physics and chemistry constantly throw unexpected structure and marvelous little symmetries in our faces, in even a cursory look? Wouldn't a true, unbiased anthropic cauldron simply toss out a universe that works fine for life, but shows no unnecessary correlations or symmetries between the resulting diverse components of its physics? Such patterns and correlations would after all represent an unnecessary, irrational, and mechanistically inexplicable "extra effort" on the part of the anthropic cauldron, an effort that goes far beyond what is needed simply to enable life. If you own a true anthropic cauldron, Occam's razor says "why bother?" with anything more in the product.

Or stated another way: I have no problem with using anthropic ideas to adjust the ratio between two tightly meshed gears, but I have a lot of trouble with using it to create the gears themselves. Nearly every finding in physics seems to be shouting at us that the bones and tendons of the universe arise from complex permutations and various degrees of breaking of symmetries, with many of details of those symmetries and their permutations being being captured at least partially in that marvelous work called the Standard Model.

So, my real message on this issue is a simple one: Extreme applications of otherwise good ideas tend to be wrong, often rather spectacularly so. Exclusion of extremes is a nicely general principle that applies to a very wide range of phenomena, and I just can't see any good reason why the anthropic principle should get a waiver from it.

An experimentalist's view:

I do not see the need to search further for why the three quarks add up to the electron charge than that given by the group structure of the Standard Model. The SM is very successful in organizing into beautiful symmetries the particle and resonances data gathered the last sixty years or so. There is no experimental reason to assume further layers of compositness defining a "deeper" group structure from which the "measured" SU(3)xSU(2)xU(1) should emerge. It will just introduce a lower level of unnecessary complexity.

If what intrigues you is the unit one, after all we can always say the down quark has charge -1, the up quark 2 and the electron -3. The group symmetries are the same and we will have a generic unit 1 .

I would say the deepest reason is anomaly cancellation. If the charge of proton and electron were not the same (even 1 in 1000000!) then the current conservation in the standard model wouldn't be fulfilled due to anomalies.

They say that anomalies have topological roots.

Given the fact that electric charge is quantized in the EM sector (Charge Quantization, Compactness of the Gauge Group, and Flux Quantization), the aforementioned anomaly cancellation technique implies the charge quantization for hadrons that you demanded.

For a more detailed explanation take a look at these pages of Schwartz QFT book: 633 and 634.