Merriam-Webster's Definition of American: (n) A native or citizen of the United States.

While you're right that the 'Americas' include North and South America, an American is a resident of the United States, much in the same way a Brazilian is a resident of Brazil, a Canadian is a resident of Canada, and a Cuban is a resident of Cuba.

So KLowD9x is not technically right, as the explanation above is the technical explanation.

edit to respond to edit:

Europe has Europeans. Check.

Asia has Asians. Check, Check.

Australia is still just the one country, but they're Australians. Check*3

'America' is not a continent like Europe, Asia, and Australia. America is an abbreviated version of the United States of America, and is the formally accepted and recognized English nomenclature having been used in several international documents as such.

However, to go off your line of thinking (Continent -> Continent+Demonymn) here is the logical conclusion:

North America (Mexico, Canada, U.S., Cuba) -> North American

South America -> South American.