At the very first time when the US approached to the East Asia, different people thought up different ways to write "America" in Chinese characters, which included 亜米利加, 亜墨利加, 米利堅(American), or 亜美利加. Later on, somehow Chinese and Korean people chose 亜美利加, whereas Japanese people chose 亜米利加/米利堅. One of the earliest example of using "米" for America is Kanagawa treaty(America-Japan Treaty of Amity and Friendship) in 1854. But I couldn't find a clear answer for "why" they chose that. Maybe there is no special reason because it's just customary thing. Reference http://www.ss.iij4u.or.jp/~ana/chimei.htm

Excuse for writing this in English because I can't explain it in Chinese but I only understand what you asked. This word 米国 is shorten word for {亜米利加} and 国 which means country in both Japanese and Chinese is added to the end. The pronunciation is different when it was written as {米国｝ so it's probably awkward to see this word as like "world of rice" but it's only made by the pronunciation.

亜米利加（ a me ri ka ）　⇒　米　 亜米利加　合衆国　⇒　米　国

亞米利加合衆國 ⇒ 亞國(can be mistaken for Asian countries) ⇒ 米國 We Koreans call the US 'sal-kook' as a joke, means rice country. It sounds more like rice soup to me though.

アメリカの米だよ。