Question: Think about it: capitalism has brought so much prosperity to so many people.

Yet no country claims our national ideology is capitalism - none that I know of.

Americans can proudly claim they are capitalists. Well. First. Not so much. And then, it's not technically the American national ideology.

There are "Socialist states of...". There are "Islamic states of...". Then there is "Democratic republic of..."

What's funny is that countries that claim they are democratic are usually not democratic either. North Korea is called the Democratic People's Republic of Korea. But that would be another question.

Why does no country ever claim capitalism as their state ideology?

Why is there no "Capitalist Republic of America"?

Answer: 
There's two kinds of antonyms: opposites (left and right) and present/absent (light and dark, where dark describes the absence of light).

Capitalism is basically the default economic system. It has both good and bad aspects to it. Even in total anarchy, you'll find Capitalism in some form. Nobody establishes Capitalism. The reason it is the default is that it tends to be self-leveling in managing supply and demand (i.e. [market forces](https://www.investopedia.com/ask/answers/012815/how-does-invisible-hand-affect-capitalist-economy.asp)).

Communism, for contrast, is a **political philosophy** that says the government should control the economy, with the theory being that Capitalism is inherently unfair. The goal of Communism is, more or less, to use the power of government to remove Capitalism (in other words, create the absence of Capitalism). It does this by attempting to manage supply and demand with direct regulatory power (i.e. the state owns everything). This can, however, produce [disastrous results](https://en.wikipedia.org/wiki/Holodomor).

Most modern states regulate Capitalism in some form. Those that want to aim more for the ideals espoused by Communism (i.e. Socialism), without the disastrous side effects, have states limit how much control they exert over various areas of the economy.

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> **NOTE** This is oversimplified for the sake of argument. The question pertains to why Capitalism isn't a political system. Don't get bogged down in the precise economic terms please.
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Apart from all the other answers, no nation should call itself "Capitalist Republic of ..." because there isn't currently any nation which is purely capitalistic. We mostly live in **mixed** economic systems, a mix of capitalistic and socialist rules, so we don't starve to death if we are poor or have to redistribute all our wealth if we are rich.