We say "U.S. citizen", but why can't we say "China citizen"? Or can we?

Is the word "U.S." an adjective or a noun modifier in this case? It seems to me that it is an adjective that goes before citizen, because we say "Russian citizen", "Chinese citizen", etc. But if it's a noun modifier, then why can't "China" be a noun modifier, too? Although "China citizen" sounds wrong to me, "(People's) Republic of China citizen" doesn't sound as bad. I know I could always write "citizen of China" and in general, "citizen of [country]", but I'm just curious to know.

Adjectives are the prototypical modifiers of Noun Phrases. Generally speaking we kind of need a specific reason to use a Noun to modify another Noun when an adjective is available. There are many such reasons, but a common one is when there is simply no adjectival alternative - in other words when it simply doesn't exist. Because there is no reason not to use the adjective Chinese when modifying nouns, we do! We should note that by and large there's no difference between modifying the Noun citizen or any other Noun used in formal English:

• Chinese citizens, officials, economics, food regulations

When the name of the country is a Noun which has no readily accepted Adjective as a counterpart, then, as Jlovegren says, we use that Noun as a Modifier in the Noun Phrase when we need to modify another Noun:

• the Myanmar people
• Benin culture

When it comes to countries whose names are already compounds, then it depends on the head word in the Noun Phrase that constitutes the name. If that head word has no Adjective counterpart we will use the name as a Noun modifier. In the United Arab Emirates for example, Emirates is the head Noun in the Noun Phrase. We have no Adjective for Emirates (Emiratian or anything like that), so we have to use the whole name as a Noun modifier. The same goes for the United States. There is no Adjective for States. So we have:

• United States citizens
• the United Arab Emirates economy

This can be contrasted with, for example South Korea. Korea is the head of the Noun Phrase here. We have an adjective relating to Korea, namely Korean, so the following is perfectly fine:

• the South Korean food industry

Similarly we have an adjective corresponding to Britain in Great Britain, so we can talk, for example, of:

• the (Great) British public

Other examples are:

• the Saint Lucian president, the South African Government, the Saudi Arabian peninsula

... and so forth.

This won't work of course if the head noun is just a generic word like Republic. In fact it won't work if there is any phrasal genitive, X of Y construction such as we often find with Republic. Consider:

• a United States of American citizen (wrong)
• a People's Republic of Chinese statesman (wrong)
• a Dominican Republican citizen (wrong)

Instead, when such compounds are used we find:

• United States of America citizens
• Republic of China citizens
• Dominican Republic citizens

Very often we have more than one name for a country. Often one is longer and more formal than the other. If there is an adjective for the shorter name we will use that as a modifier accordingly:

• Vatican officials

However if we use the compound name, then as described above it will depend on the head Noun in the Noun Phrase. So in instances where the head Noun has no adjectival counterpart we will find examples such as:

• Vatican City State officials

Similarly, countries primarily known by their abbreviations will also have the abbreviations used as modifiers in Noun Phrases, because there are no adjectives (or at least well-known ones) relating to the names of letters! So we have the British government, but also:

• the UK, US, and UEA governments

This goes for abbreviations in general, so we commonly find phrases like:

• WWF supporters, IBM contractors, FIFA officials, IMF spokesmen

Hope this is helpful!

Why can't you say China citizen?

Because you can already say "Chinese citizen". The phenomenon is called "blocking" in linguistics. You don't say something because there's already something else that you do say. For example, you you don't say "gloriosity" ( < glorious) because you can already say glory, even though by analogy with viscous>viscosity, you would think it's a reasonable thing to say.

By analogy with U.S. citizen, you think you can say China citizen, but Chinese citizen blocks it. U.S. citizen is different either because it predates American citizen or it means something different. e.g., it's shorthand for the legal term "citizen of the United States" (see below).

Also, United States doesn't have a corresponding preposed adjectival demonym, but China does. My hypothesis is that countries where there is no such form, or you don't know it, you treat like "US citizen": Vanuatu citizen, UAE citizen, Papua New Guinea citizen. English speakers, do these sound better than "China citizen"?

If you know the preoposed adjectival form, then it blocks the N-N compound: Czech, Russian, Soviet, etc.

Why can "U.S." act to modify a following noun?

Another guess: it comes from journalism. There are newspaper headlines with phrases like "U.S. troops", "Boston man", "Philadelphia lawyer", etc. which were used in newspaper headlines, then in radio broadcasts, then in television broadcasts.

But English can productively form N-N compounds, so when someone says "U.S. citizen", a native speaker can make a good guess about what it means, even if he/she hasn't heard it before. E.g., a freshly-coined, highly embedded series of N-N compounds such as dog food factory management regulations will be understood by most English speakers.

And U.S. citizen itself...

I think this is shorthand for "Citizen of the United States" invented by U.S. lawmakers. The 14th amendment to the U.S. constitution refers to "Citizens of the United States". The U.S. Code (1918 U.S. Compiled Statutes) gives the following definition for U.S. Citizens as of 1918:

All persons born in the United States and not subject to any foreign power, excluding Indians not taxed, are declared to be citizens of the United States. (25 USC 3946)

The phrase "citizen of the United States" is used throughout the title on citizenship. e.g.,

Whenever any of the chiefs, warriors, or heads of families of the tribes mentioned in section twenty three hundred and ten, having filed with the clerk of the district court of the United States a declaration of his intention to become a citizen of the United States, and to dissolve all relations with any Indian tribe, two years previous thereto... (25 USC 3950)

This phrase still appears in the modern-day editions of the U.S. Code: i.e.,

The following shall be nationals and citizens of the United States at birth... (8 USC 1401)

But the wording "United States Citizen" can also be found in modern editions:

...a single offsense of simple possession of 30 grams or less of marihuana, who (A) is the spouse or child, of a United States citizen, or of an alien lawfully admitted... 8 USC 1182 (1976 ed.)

...and has in writing irrevocably released the child for emigration and adoption; who has been adopted abroad by a United States citizen and spouse jointly... 8 USC 1101 (2000 ed.)

as well as earlier versions:

That it shall be the duty of immigration officials to record the following information...Name, age, and sex; ... whether able to read or write; nationality; country of birth; country of which citizen or subject; race; last permanent residence in the United States; intended future permanent residence; and time and port of last arrival in the United States; and if a United States citizen, whether native born or naturalized. (32 Barnes' Federal Code 3711 ; 1919)

Interestingly, the same passage of the 1919 version of the code (sec. 3664) uses the phrase "Chinese subject" rather than Chinese citizen:

The provisions of this act shall apply to all subjects of China and Chinese, whether subjects of China or any other foreign power; and the words Chinese laborers, wherever used in this act shall be construed to mean both skilled and unskilled laborers and Chinese employed in mining.

What about "American Citizen"? This word made its way into Title 25 due to a law passed March 2, 1907 (34 Stat. 1229), which uses both terms with no apparent distinction in meaning: