How does it not? I guess companies can trade their stocks at foreign exchanges, but over time that should cease. Assuming that productivity per person becomes more than half what it is in the US and that Chinese companies will trade their stocks in a domestic exchange it does indeed seem inevitable. Those aren't very wild assumptions.
"over time that should cease" why? companies are globally owned, listing is largely about which accounting rules and disclosure and ownership rules you follow, no need for a home bias in the long run.
Look at London for an example of a stock exchange largely full of international companies.
By that line of reasoning, India's stock market should be the same size as China's, and both China and India should have a stock market that is 4 times bigger than the USA. But that is not going to happen.
As developing economies 'develop' their per-capita GDP approaches that of developed economies. Therefore, as this happens, the size of an economy becomes more proportional to the population size.
Therefore over the very long term that absolutely should happen.
Not quite. Your position rests on the premise that China can bring its population up to developed status (something on par with the US, Britain, France, Germany, Japan etc), ditto for India. That is very unlikely to be feasible, given the truly massive manufacturing, consumption, exports, energy, resources, etc. that it would require to boost China's GDP to ~2x the size of the entire rest of the world just to bring their median income up to that of the US.
That is not going to happen soon. It could happen in the future, for China at least. India is a poorer country, but both have histories of being industrious and productive, so it really shouldn't be surprising if they ascend again over time.
http://www.internetworldstats.com/stats8.htm
So, I don't see why you would think that's inevitable.