Some can, some can't. I do think that Democracy is more sensitive to national separatism than other forms of government. The great multiethnic empires of history weren't democratic. The Chinese have held together a multiethnic society for 3,000 years without holding a vote.
The Northern coalition in the Afghanistan civil war was composed of many ethnic groups fighting together against the Pashtun south. There is enough common ground between the groups of the north that a two state solution may suffice.
Nationalism is a powerful force that has great appeal to human psychology. Every successful multi-ethnic, multi-religious, or multi-lingual empire has to address it. Even the Canadians have struck an uneasy balance between the French speaking Quebecois and the rest of the country. Separation was a close thing.
And nationalism doesn't skip Europe. Try to find Czechoslovakia or Yugoslavia on a map. And soon Spain might be joining them in the historical deadpool if the Catalan independence movement gets its way.