Would a Grexit from the Eurozone create any strategic problems? Absolutely. If other Eurozone countries force Greece out of the currency union, we should expect it to have a deeply damaging effect on the NATO alliance, which remains the crucial lever by which the United States organizes security contributions from European countries. Greece has always been a balky, reluctant member of NATO, so it is tempting to believe a rupture in the Eurozone would be followed by an embittered Greece withdrawing also from NATO, relieving us of the hassle of dealing with them—they would become just one more poor, prideful country that has little effect on the international order. But if Greece were to leave NATO, the West would lose their significant contribution as a fighting force, access to bases instrumental for power projection in the Mediterranean Sea and Middle East, and use of a large merchant shipping fleet.
Moreover, a Greece stripped of NATO’s protection (even if it should choose that outcome) and Europe’s financial underwriting would likely search for alternative providers. The Tsipras government early on in its financial crisis flirted with a Russian option; Russia’s resentment of the West, growing confidence, and a rising price for oil (occasioned by increased skittishness in financial markets from Greece leaving the Eurozone and NATO) could make the prospect more enticing. China, too, might see advantage in securing Mediterranean ports or a foothold and listening posts in Europe.
Even if Greece did not flout allegiances with non-Western countries, it could create strategic problems for Europe by simply refusing to cooperate. Outside the Schengen compact, Greece would have no obligation to assist in managing immigration. Outside the Eurozone, Greece would have no obligation to govern itself transparently or well, no requirement to open its books for inspection or implement anti-corruption measures. It would become less European and more Balkan, which would make Europe less stable and less safe.
Operationally, too, Grexit could be problematic. If Greece remained in NATO, the economic devastation would surely impede Greece’s ability to provide security both for itself and to others. Greece leaving NATO would incentivize Turkey to test claims in the Aegean, raising the prospect of conflict in which NATO countries would be obligated on Turkey’s side against Greece. At a minimum this would result in intensive mediation distracting from other, higher priorities for the alliance; further up the scale, it could see other countries joining Greece’s side to array against NATO.