Heading south against what?  The value of a currency is relative, it is measured in the value of other currencies.  Basically what you are suggesting is that the Euro will go down in relation to the dollar and the yen.  There's no logical reason to really think peak oil would be a good event in the U.S.  Although we still produce some of our own oil, we also use more oil than anyone and have as much or more dependence on it than any other country.  
But the US dollar is the currency of denomination, and of reserve.

It's easy to see a world where the US dollar is 40% devalued (such would be logical, and should have happened, bar domestic Chinese and other South East Asian policies, which prefer to hold down their domestic currencies by holding USD assets).

But a really big devaluation beyond that, when roughly half the world's economy is denominated in dollars, is hard to see.

Also Europe is in a relatively worse position in a PO scenario.  Yes we consume less, but we import 80% of our energy (coal mines in eastern Europe, and some renewables, plus of course the North Sea, but that is about it).

The US by contrast is the world's largest or second largest coal producer. Plus it has access to Canadian tar sands, Arctic gas etc.  And it has big biomass resources (wood for heating etc.).

Another factor (the same for CO2 emissions) is that the US would find it relatively easy to reduce energy consumption-- simply switching to diesel powered cars, ditching SUVs etc.  By and large, Europe has already made those sacrifices.

I should add that I don't think the euro in its current form is a long run currency.

Eventually Italy will find a way to exit, and Greece also-- both countries are effectively having their economies strangled by deflation.

I don't think many of the EU25 countries that are not in the Euro (UK, Denmark, Hungary, Poland etc.) are likely to rush to join it-- the ones who would benefit (Hungary) are precisely the ones in the worst fiscal position and break the ECB rules.

The euro will revert to what it always was: a Franco-German currency union, with Benelux (and maybe the Irish and the Spanish) along for the ride.

Except the euro does bring a lot of benefits for the integrated economies in turn - a manufacturer no longer has to deal with the overhead of currency transactions/risks.

And by now, the banks have gotten used to living without the money they used to be able extract from intra-EU trade, which means that EU wide companies - oh, like Unilever, or VW, or Barilla - find a lot of benefit in the euro.

Whether the euro has a longer term future than the dollar is open, but quite honestly, for the economies within the euro, the negative side is incresinly harder to firmly determine.