I read that the vast majority of rice farming done in Europe is in countries on the Mediterranean because it provides good conditions for it. It seems like an empire that controlled the Mediterranean could make use of how productive it is compared to wheat.
It was not.
The Romans did consume a number of grain and grain products. Rice was not counted among these. Nor were many other grains that are popular nowadays (or were popular in other geographic areas at times contemporary with the Romans); "Maize, sorghum, rice, oats, rye and the millets, were either unknown, or not grown on a large scale in the Mediterranean world of antiquity" [1, pg. 261].
Jasny's "Competition Among Grains in Classical Antiquity" provides the clearest answer to why the Romans ignored these other grains. Some of them, like Corn, were simply absent from the old world, and it would take colonialism and millennia for it to be introduced. Others were poorly suited to the climate; Rye is convenient as a winter-resistant crop -- a relatively useless trait in the balmy Mediterranean winters. But for most of the others, including rice, the issue was one of irrigation [2].
Rice was certainly known to the Romans. It had been "introduced in Babylonia many centuries before [the Classical Era] and from there penetrated farther west" [2, pg. 748]. However, it took a great increase in "permanently irrigated land" before it was widely cultivated, which did not take place until "well after the classical period had passed" [2, pg. 748].
Rice was consumed to some extent in Rome, largely as the result of trade with "lower Syria and Bactria" [3, pg. 240]. It would have been considered another rare oriental good, used as "broth or cake, and when kneaded with bone-meal [...] by ladies for preserving the smoothness of skin" [3, pg. 240].
There are some limited mentions of Rice in Roman writings. Strabo mentions it in his Geography, when he speaks of India in book 15. He mentions it as one of many crops the Indians sow in the rainy season, and refers to Aristobulus to describe its growing conditions, the plant itself, and the timing of its harvest. He notes that, at the time, rice "[was grown] in Bactriana, Babylonia, Susis, and in the Lower Syria" [4, book 15, chapter 1].
There is also a brief mention of rice in Horace's Satires. A physician, brought to address the "prodigious lethargy" of one Opimius, exhorts him to "take this ptisan of rice" [5, book 2, poem 3]. Ptisan refers to ptisanarium, a sort of rice gruel. The physician mentions that it cost "A trifle [...] Eight asses" [5, book 2, poem 3]. It's unclear how much rice would have been used in that gruel, but this example provides both another use of the grain and perhaps a rough estimate of its value.
Rickman, Geoffrey E. "The grain trade under the Roman Empire." Memoirs of the American Academy in Rome 36 (1980): 261-275.
Jasny, Naum. "Competition among grains in classical antiquity." The American Historical Review (1942): 747-764.
Sagar, Krishna Chandra. Foreign influence on ancient India. Northern Book Centre, 1992.
Hamilton, H. C., and W. Falconer. "The Geography of Strabo. Literally translated, with notes, in three volumes." George Bells & Sons, London (1903).
Buckley, Theodore Alois, ed. The works of Horace. G. Bell & sons, 1885.