Yes, extreme violence+ethnic cleansing efforts can lead to the emergence of stable national identities. This process could be applied on a global scale to give you the results you are looking for.
A politic and governing system founded on the removal of others is gonna either fail because it runs out of others to remove, or, more likely, will keep making others of its own people in order to justify its existence
No one is pure. There is no such thing as a pure ethnicity. We're not even all human, with some of us having neanderthal DNA, so creating a society designed to accept only one kind of person is effectively impossible, as there will always be hairs to split--And split those hairs they will when you found a society on the premise of blaming an other for your problems
Besides, even ethnic purity is no guarantee for stability. Countercultures exist, and revolutions happen for things besides racial means. There is still class warfare, not to mention the every-splintering world of spiritual beliefs