Rice was first domesticated in China around 10,000 years ago and very quickly spread through East Asia and the Indian subcontinent. The two main varieties of rice (Oryza sativa) today are japonica, which despite the name is from China and not Japan, and indica from India. There was a separate domestication event around 2,000 years ago in West Africa, but this variety is only grown in very small amounts and is confined to West Africa.
The reason why it did not spread very far into Europe is because older varieties of rice require high moisture and a relatively hot climate, which was uncommon in Europe. Although rice can be grown in some places in southern Europe (Italy and Spain), the climate is less suitable and the varieties grown there have only adapted recently.
Europeans have known about rice at least since the time of Alexander. The Romans imported some rice, but it was in small amounts. The closest source for them was Egypt, which grew a variety of West African rice, but it was low yield and hard to grow in Egypt’s climate. They also imported some from Asia, but that was much farther away and expensive.
Rice was not grown in Europe until the 10th century, when the Arabs introduced it to Spain and Sicily. Arabs had themselves only been introduced to rice a few centuries earlier during the Muslim conquests. In Spain, rice was grown sporadically in a few places in the south, but it seems to have fallen out of favor during the Reconquista, and has only been reintroduced in modern times. From Sicily, rice spread to the Italian peninsula, and from around the 15th century, there has been a rice growing region along the Po river in Lombardy.
The new world did not know rice until it was brought over in the Colombian Exchange. The climate was suitable for rice cultivation in parts of south and central America, and also along the Mississippi river in the southern US. Initially, European colonists were not very interested in growing rice because it wasn’t part of European cuisine, but several factors changed that. One was the annual flooding of several rivers which favored rice because rice can tolerate flooding while wheat can’t. Another was the fact that most of the slaves imported came from West Africa, where they were already familiar with growing rice, but didn’t grow wheat. Their rice farming expertise was handy for the European colonists.
The reason why rice is more popular in Brazil and the Caribbean than in the US is probably because of the higher proportion of West African slaves relative to European colonists. Their original cuisine had more rice to begin with, compared to Europeans. However, these differences are only relative. The vast majority of the rice produced and eaten in the world is still in Asia. China and India are the largest rice producers and consumers.
In more recent times, several new varieties of rice have been developed that are more resistant to dry conditions. This has increased the range where rice can be grown. The global economy and cheap transportation has also enabled countries such as the US to grow rice for export, even if they don’t consume much internally. But 90% of the world’s rice is still grown in Asia.