"American-born" is the common phrase.
Firstly, it's best to think of it as as a set phrase that refers to citizenship, rather than country of birth. While you might be understood if you say "America-born," the phrase will sound odd to a native speaker, and most people will simply assume that you meant to say "American-born" and attribute it to a fluency error.
Secondly, a searchable term for the second part of compound words like
American-born is a hyphenated suffix. Hyphens have several uses, one of which is to connect modifying words, especially when needed to clarify or separate words that could be parsed incorrectly. For example, consider the example ofsentence fragments below:
American born abroad.
Someone who is currently an American citizen but who was born outside the USA. They may or may not have been an American citizen at birth.
A person (or people, plural; the fragment is ambiguous without context) currently living or traveling outside the United States. This person was an American citizen at birth.
In practical usage, the hyphen here syntactically allows the grammatical but non-colloquial phrase
born as an American to be converted from a verb plus modifying clause into a single compound word. In either case, American is actually eliding "American citizen." Therefore, a native speaker would generally understand American-born to mean "born (as) an American citizen."
Because English is a satellite-framing language, phrases that denote motion or manner often require or imply a preposition or prepositional phrase. If it helps, you can think of American-born as meaning "born into American citizenship."
"America-born" is not a common phrase.
While probably grammatical for certain usages, the phrase "America-born" is simply not one that you are likely to hear or read. If you wanted to say that a non-US citizen was born in the USA, you would say something like
A <nationality> born in the US. The nationality would generally be expressed as an adjective like Chinese, Canadian, or Dutch (implying the noun citizen).
There are some exceptions. For example, you're more likely to hear "An Irishman/Scotsman born in the US" than something more contorted like "An Irish UK citizen born in the USA." Nevertheless, such phrases are still generally understood to reflect citizenship rather than cultural or racial identity. When talking about identity, the phrase is generally
<adjective>-American such as Irish-American, African-American, and so forth.