This is mainly a question of environmental history, but also societal organisation - rice cultivation requires both the right kind of climate, and intensive organised labour. However, you have a few misconceptions about European rice that I want to clear up.

Some European and Middle Eastern cultures did cultivate rice, though not the same variety as what's grown in East Asia. Domesticated rice can be informally split into two 'groups', indica (long grain and non-glutinous) and japonica (short grain and sticky). While this is a bit outside of my area of understanding, some varieties of indica rice were known to the Romans and Greeks likely through the eastern trade. There's evidence of rice existing in the Greco-Roman world, but it wasn't grown in Europe in significant amounts until it was introduced by the Arab/Muslim conquest of Iberia and Sicily in the 8th century, where it remains a staple part of southern Iberian cuisine - and their Latin American colonies - to this day. As for why rice didn't take off everywhere in Europe, you are correct in your suspicions that climate plays a part.

Rice requires a hot and humid climate, with a lot of prolonged sunshine and moisture. It is a semi-aquatic species that requires constant irrigation. If you can fulfill these needs, then it is a very hardy crop that can grow practically anywhere, and handle a surprising amount of disturbance. As such, the natural range of rice species is in tropical marshes and wetlands, or along temperate river banks. You will find a significant overlap between the human societies that settled these kinds of environments and the cultivation of rice. It's simply the best suited crop for the environment it's in. In comparison, wheat handles cold climates better, and requires less water. In a lot of places, the split between wheat and rice cultivation is simply climatological. Referring to your question about China, you will see that the split between rice cultivation and wheat cultivation matches near perfectly with temperature and precipitation charts. While these are modern statistics, the historical range of rice cultivation hasn't really changed much over the millennia, except a slow northward expansion as the land was terraformed and canals were dug.

To answer your question about monsoons, it's not necessary for rice, but the heat and humidity that comes with being monsoon territory helps a lot - rice is also a hardy species that can survive monsoon conditions, unlike some other common types of crops. Southern China, eastern India, and tropical Pacific Asia have absurdly good conditions for growing rice. The year round heat and humidity allowed for double or even triple cropping of rice, a feat that few other places in the world could match prior to industrial fertiliser.

Climate, however, isn't the only answer. There are parts of India that could grow rice, but instead cultivate wheat. In Modern times you can find rice paddies in Italy, France, Australia, and even Siberia. Large parts of the South in the US and even the Great Lakes have prime conditions for growing rice, yet wheat is the dominant crop. So why is this the case? At this point, we have to step away from climatology and look more at the human societies cultivating the crops. This could be a massive post on it's own, but for the sake of brevity I can give you a short rundown. The conventional arrangement for the cultivation of rice is the rice paddy, with variations across all the world but sharing the common characteristic of being a flooded field. It's a terraformed environment designed to replicate those perfect wetland conditions, but at the cost of suitability for other types of crops. Because of this, rice is often a monoculture and monocrop. Not many other crops can be grown alongside rice, and most fields couldn't be easily converted back and forth between rice and other forms of crops. The establishment of rice paddies is a significant undertaking of labour that requires hydrological engineering knowledge for the terraforming of land and the digging of irrigation ditches and canals. Prior to mechanisation, building rice paddies - especially in the mountains - was legitimately a generational effort, and some rice paddies in China like the famous Honghe Hani Rice Terraces are over a thousand years old, with continuous cultivation, maintenance, and expansion over dozens of generations.

At the risk of making sweeping generalisations, not every society had the capacity or desire to commit to that level of labour for the sake of rice, especially if it overlapped with more culturally familiar forms of agriculture. For example, the indigenous Ojibwe and Dakota of the Great Lakes in North America cultivated manoomin/psin (wild rice) for thousands of years in their own way, but most of that managed ecosystem was lost to European colonisation who either didn't recognise it as a form of cultivation, or preferred to supplant it with their traditional/colonial form of agriculture.

Some References

• Nagai, T., Makino, A. (2009). Differences Between Rice and Wheat in Temperature Responses of Photosynthesis and Plant Growth.

• Fuller, D. (2011). Pathways to Asian Civilizations: Tracing the Origins and Spread of Rice and Rice Cultures.

• Talhelm, T., Zhang, X., Oishi, S., Shimin, C., Duan, D., Lan, X., & Kitayama, S. (2014). Large-Scale Psychological Differences Within China Explained by Rice Versus Wheat Agriculture.

• Hannaway, D., Hannaway, K., Daly, C., Chapman, D. (2003). Using new technologies to map the suitability of forage plants.

• Zhang, S., Zhang, D., Li, J., Pei, Q. (2020). Secular temperature variations and the spatial disparities of war in historical China.