By my count, there have only been three countries (possibly four) that claimed to be founded on ideas—rather than a particular religion, culture, or ethnicity—and which believed these ideas were objective, universal, and needed to be spread across the world.
The first and most obvious is probably the United States, for reasons most of us know.
Coming shortly afterward was France, which in some ways took things even further—mostly because it was going up against a thousand years of entrenched monarchical traditions, in a continent full of hostile monarchies. For example, to this day, the French constitution forbids the government from collecting data on race, religion, or national origin to preserve the idea that all people are equal in their status as citizens (and that citizenship is not contingent on such things).
Finally, there the Soviet Union, which tried to forge an entirely new nonethnic identity (Soviet) based around an entirely new idea (communism), upon a society that had previous been deeply religious, multiethnic, and largely feudal. Soviet ideologues even devised the idea of the “New Soviet Person”—someone defined by traits and virtues that transcended nationality, language, etc. We all know how well that turned out.
Of course, all three countries did not live up their ideals in practice, with the Soviets failing altogether. “True” Americans were (and to many people remain) narrowly idealized as white Anglo-Saxon Protestants, so that even black Protestants or white Catholics were, in different ways, seen as suspect. Both France and the Soviet Union gave greater privileges to white French and Russian speakers, respectively, etc.
But these are still the only countries that had at least the pretense of being universalist and idealist in their national identity (at least to my mind).
(Switzerland comes close, uniting four different ethnic and linguistic groups, and several religious sects, on the basis of a shared alpine identity and a commitment to constitutional federalism. But it never developed anything close to the manifest destiny of the U.S., the French Republic, and Soviet Russia.)