Even if we assume that a nation cannot exist without borders (itself a contestable claim because many nations have historically had unclear or contested boundaries), it does not follow that the maintenance of borders requires immigration restrictions. In reality, borders have a wide range of other functions, besides regulating immigration. For example, they define the territory within which a given government’s laws are binding, and also the land area within which it may deploy its armed forces without getting permission from other governments. If all immigration restrictions were abolished tomorrow, borders could readily continue to facilitate these and other purposes. A nation that doesn’t exclude peaceful migrants can still bar invading armies.
The history of the United States also shows that borders – and nations – can exist without immigration restrictions. Until the Chinese Exclusion Act of 1882, the federal government did not forbid voluntary immigration. Indeed, the original meaning of the Constitution did not give Congress the power to do so, allowing it to restrict eligibility for citizenship, but not to forbid migration. Some state governments had laws excluding immigrants, but not the federal government (and migrants excluded by one state could still potentially enter through another).
If we take Trump’s theory (and others like it) seriously, the Declaration of Independence did not make the United States a nation because it did not establish any immigration restrictions. Even worse, it condemned George III for “obstructing the laws for naturalization of foreigners [and] refusing to pass others to encourage their migration hither.” Instead of celebrating Independence Day on July 4, we should commemorate the enactment of the Chinese Exclusion Act. Jefferson Davis and his friends need not have taken the trouble of trying to secede from the United States in 1861. They should instead have argued that it simply did not exist in the first place.
Even today, some nations, such as Argentina, do not restrict immigration. Few would argue that Argentina is not a real nation, that it has no borders, or that it somehow ceased to exist when it adopted a virtual open borders policy towards migrants in 2004.
The debate over immigration policy raises a number of genuinely complex issues regarding the economic, political, and cultural effects of migration, and the extent to which it is morally permissible to make immigration policy without considering the freedom and well-being of would-be immigrants themselves. There are restrictionist arguments that deserve serious consideration, such as claims that immigration might create dangerous “political externalities” that reduce the quality of public policy. But the assertion that we must restrict immigration because nations cannot exist without borders isn’t one of them.
UPDATE: I should emphasize that this post addresses only the claim that the inherent nature of nations requires them to impose immigration restrictions. It does not consider claims that immigration might have negative effects on nations (without causing them to cease existing), or even exclude the possibility that there might be extreme situations where the negative effects of immigration are so great that the nation in question collapses. The validity of such claims may vary from case to case, and certainly is not inherent in the definition of what it means to be a nation. Moreover, even where immigration causes some negative effects, it is often possible to mitigate or eliminate them without restricting migration itself (which is not true of claims that immigration restrictions are inherently necessary for the existence of a nation).
UPDATE #2: It is worth noting that claiming that nations can’t exist without immigration restrictions is not the same thing as claiming they can’t exist without restricting freedom of movement in some way. There are a variety of situations where restrictions on the movements of foreigners might be justified in cases where we would also be justified in restricting the movement of comparably situated native-born residents. For example, governments are often justified in restricting the movements of terrorists, violent criminals, spies, and carriers of contagious diseases, regardless of their nationality. Immigration restrictions, by contrast, forbid entry simply on the basis of where the person in question was born or resides.