THAT would have an effect on the USD eventually. But it will also be damaging to the Europeon economy, making their goods too expensive to import. So it works both ways. A weak USD would be more competitive and benefit exporters of American goods, while it hurts importers of Europeon goods, reduces our buying power. An example of the pros and cons can be seen in the Canadian dollar market right now, where the canuck buck is near par with the USD. It's damaging to exporters of Canadian goods into the USA, because they aren't as competitive with American manufacurers as they were when the canuk buck was around 70 cents. On the otherhand, they are making a killing on oil and gas.
The net effect of the weak dollar is a higher price at the pump. Yes, there are many other factors, but the dollars fluctuation is not "irrelevant" in a world traded commodity.