No country is actually going to go "extinct"
I'm not talking about countries, but nations. It's geographical ethnic and cultural in nature.
Thought experiment: Let's look at Germany. There live Germans with their values culture and events. There is immigration from middle east. For now they are minority. If they reach majority, they can impose their values on Germany at large. For example Oktoberfest wouldn't go too well if alcohol gets banned.
Plus, in the U.S., we have mass immigration and automation to make up for native born birth rates declining in the future.
Think about how culturally and politically relevant are native Americans. Considering how much Republicans like to say America is Christian nation, my guess is that they aren't all that relevant. Decline of native Americans have different reasons but outcome will be same.
And Europe will be more affected by this as USA doesn't really have single ethnic identity, while most countries in Europe do.