IMO no. Others mention the network effect, and while it will certainly be a factor, I don't think it will be anywhere near strong enough to make the monopoly plan work.
The network effect for Uber is really local - at the city level. An "Uber, but only in Austin" service can easily compete with a national-level Uber, since the vast majority of cab rides are taken within a user's home city.
You need a critical mass of cars/drivers to present real competition in this space, but you don't need to do it at national, or even state-level scale. A small upstart that can achieve critical mass in a single city can present real threat to Uber in that city.
The network effect of cabs between cities is so little that while a lot of consolidation of cab companies have occurred in each city, until Uber there was never a major cross-city network. I'm not convinced that a unified worldwide fleet has a significant competitive advantage. It'd be useful for tourists, maybe.
And this is already happening - Juno here in NYC is gaining real traction by charging drivers less, getting drivers to evangelize to each other. Uber won't be toppled by Lyft or any other megalithic ride hailing company, it will more likely be toppled by hundreds of smaller, geographically narrow companies.