First off not the EU and thus NATO would now get involved in the conflict. However NATO would join the Swedish and Finnish in any war regardless considering they are de facto Western countries.
The EU, I guess would. But would the US join with is NATO allies to fight Russia's action against a non-NATO state? I'm not sure the US public would be in support of going to war with Russia over Sweden. For a NATO ally, we have treaty obligations. But to risk WWIII over Sweden? Maybe the Russians think the US won't act and they can roll over Sweden, brush off Finland, annex and resolve the conflict before the rest of EU acts while the US bickers with itself on the side.
Then with Russia in control of the Baltic Sea, reinforcing the Baltic states would be impossible.