Classic (as in what we use today) democracy becomes implausable because it is about voting for policies or people, rather than overal trends. So what you get is people voting for whoever is the flashiest or with whom they identify the most with, or policies that pander to them or against people they do not like.
Democracy that would be viable is a more bottom up approach, with a federalistic system of representatives and constraints to ensure you do not get blatant majoriti-ism, while allowing all of the population to vote on broad lines. Such a system could be designed like
Thats just some ideas. Of course classic "vote once for a country-wide government" is not going to work in a big diverse country. Hell, it doesn't even work all too well in my small country, with the, ehm, non-megalopians accusing the government of only serving the needs of the people living in the megalopolis. Popularly elected people are often read never a good fit for roles of making policy on healthcare, culture, education, infrastructure. They are career politicians, people out for power or prestige, out to fullfill a political goal. The best way to run things like education is to have the educational system to regulate itself, with intervention from wider society if necessary, the same goes for pretty much everything.
Another idea I had (that does feel a bit wrong or weird, new world orderish) is to try to create more of a pan-[whatever] identity. This could be done by having mandatory conscription, stationing people randomly in places and forcing them to work and build connections with people from elsewhere in the country. The hope here would be that they would either identify or relate to these people after living there, maybe even staying there, or even better, finding spouses from other parts of the country and creating a new generation with ties to the wider country, rather than a specific ethnic group. You could also promote widespread movement of higher education students to universities in other places in the country, which has the same effect. These things can create a sense of unity among the peoples and break or at the very least impede reactionary separatist movements.
Right now, almost all the internal violence troubles plaguing both Europe and the US are a result of not integrating society better. Arab ghettos in europe breed terrorism inside them and reactionaries as a response outside them, while white-black-latino segregation creates and perpetuates racialist groups and violence in america. Every country in europe has other examples, such as mollukkers and surinamese in the netherlands and gypsies in southern europe. Reactionairy movements along ethnic lines caused the fall of Yugoslavia (with some help from the CIA) and created tension within the SU (which was enforced by economic inequality among the areas). I am not saying "do what china does" and colonize tibet with people from the majority ethnicity, but just mix it up at a faster pace than is happenening right now, and do not allow ethnicly homogenous sub-communities to form, be they yiddish, arab, african, polish, roma, armenian or whatever other ethnicity it might be. I do not advocate forced relocation, apart for breaking up ghettos inside of cities, I just advocate giving the flask a good shake periodically (conscripts, students, holidays) and letting it settle how it wants to. If only a few percentage of the population has cultural ties to multiple parts in the union, it will be less likely to break up or turn on each other.