I don't believe it. The only country that we could possibly consider having a monetary union with would be Canada since they speak the same language and have a similar culture (but not that similar because they are socialists and we are not).
In order to benefit from a monetary union the countries participating have to be highly integrated. This means there needs to be easy capital flow from one place to another and a non "sticky" labor market that will go to where the jobs are at.
The problem in the EU for instance is that while capital easily flows from one place to another labor does not. This creates a problem because if the money leaves and the labor stays what do you end up with? That's right, high unemployment.
There is a East-West boundary running across Europe where most of the skilled labor is in the north and unskilled labor in the south. This is what you don't want to happen in a monetary union.
The monetary sovereignty that the US would give up in establishing a monetary union with any other country would outwiegh any efficiencies having one currency would create.
And for that matter the US doesn't need to "compete" with the Euro. We have a better economy and better growth and lower unemployment. I'd rather have 3.5% inflation and low unemployment (4%) than 1.5% inflation and 8-10% unemployment like the EU countries.
I don't think that the sinking dollar is going to plunge the US into a recession the likes of the Great Depression or the Oil shocks. Our exports increase as a result which is good for our economy and quite frankly we need something to help us level out or dare I say begin to reduce our current account deficit.
I see it as a good thing, and I don't for one second think that the dollar is going to plunge to 55 on the index because we have a GREAT economy. We just need to increase our savings and decrease the investment/imports.