Assuming Putin or someone like him still gains power in Russia, which seems likely, probably not. Its less the existence of NATO weapons in Poland and Romania for instance than the existence of a more liberal and economically successful system of government than what Putin has in Russia that's the threat to him. Similarly during the cold war the threat the west posed was never some armed invasion but that it existed and disproved the lie that the Soviet system was the best way of doing thing, let alone the only one. The danger for Putin and for the Soviets before him is that too many of his subjects see the current ruling elite as the problem that's making their life miserable.
If you have EU expansion without NATO expansion - apart from the fact being in the EU is a probably a lot less attractive without NATO membership - then it leaves the latter in a bind when a Putin Russia starts throwing its weight about. Are France, Germany, Italy etc going to militarily support Poland, the Baltics, Romania without cover from the US, Canada or [by now] Britain? Talking here about actual fighting or at least boots on the ground. If their not willing then the assorted eastern nations have less reason to be in the EU. Does it prompt some earlier 'European' i.e. EU army/military establishment to cover this issue?