In chapter 12 of On Democracy, Dahl says that democracy requires a relatively homogenous society in which cultural conflicts, if they exist, are “weak.” Does American society meet this requirement? What type of cultural conflict, in your opinion, poses the greatest threat to American democracy? And what method (see Dahl’s discussion) offers the best hope for keeping that conflict in check?

On Democracy was written by Robert A. Dahl, a prominent expert on writing about democratic political institutions. The book can be read as a text that explains the cultural and economic underpinnings which tend to hold democracies together, and the pitfalls that tend to unravel democracies. Dahl's text is a highly informative book for those who want to understand why democracies work, why democracies sometimes fail, and how we can all preserve democratic political institutions in a time period when these institutions are under a high level of scrutiny and pressure. Most people who already understand politics but who do not know which forces work for democracy and against it would probably benefit most from reading the book.
American society essentially consists of people who believe in freedom and democracy. Although there is a mosaic of cultures, backgrounds, and religions, people in America expect their will to be carried out and respected, and this belief unites most people on some common ground. According to Dahl, that is a good starting point for preserving democracy, and I think most people would agree. But there is a high level of ideological conflict that is pervasive in American culture.
Intense partisanship and ideas about whether communist, socialist, or capitalist systems work best are being debated constantly nowadays, and this conflict seems to pose the biggest threat to American democracy, since those who subscribe to each respective system's ideas are unavoidably in opposition to each other. This division can create hostility and resentment toward groups in American society. Conflicts which concern income inequality, in particular, can foment incredible levels of turmoil, since those who believe in capitalism do not really see income inequality as a social problem. Those who believe in socialism do see it as a social problem.
Dahl makes the argument that a democratic society's preservation is tied to a healthy, functional free market economy, and that the stability of a democracy relies on limiting or resolving economic tensions that could exacerbate cultural conflict. If true, this lends support to the notion that conflicts about which system—communist, socialist, or capitalist—is best for the people in society seem to present the biggest challenge, and keeping those conflicts in check would offer the best hope for preserving democracy. At the end of the day, economic disparities and resentment are a cause for great concern, and keeping those in check offers the best hope. A method for controlling that conflict through open discussion, governmental policies, and reforms offers a good solution.

Robert A Dahl's "On Democracy," published in 1998, sets forth the assertion that democratic political institutions are more likely to develop, remain stable, and even flourish in societies that are culturally homogenous and lack the sort of sub-cultural conflicts that could weaken the society at large. These conflicts could be the result of language, politics, race and religion (among other areas). However, he does not say this is an absolute.
While many sociologists acknowledge competing and conflicting sub-cultures as an ever-present facet of American life, Dahl specifically notes that a handful of countries (the US, Canada, Switzerland, Belgium, and the Netherlands in particular) are exceptions to this notion, with democratic institutions surviving and even thriving due to a handful of interdependent processes such as forced assimilation, the shared idea of consensus, the establishment of non-partisan electoral systems, and the fear of separation for those who don't assimilate.
Dahl references the fact that most American settlers and immigrants in the early twentieth century assimilated despite vast differences in socio-economic and cultural backgrounds either voluntarily or as a result of social mechanisms like "shaming," causing them to cast off the identities of their home cultures while embracing Americanism. He notes blacks and Native Americans required greater levels of coercion to assimilate and were excluded or ostracized if they didn't comply.
According to Dahl, the greatest threat to undermining America's democratic institutions are large-scale political, economic, ideological, military, or international crises that could foment conflict between subcultures. He also notes that the principles of a democratic society are closely linked to a functional free market economy, and that the stability of a democracy relies on mitigating economic tensions that could exacerbate cultural conflict.