Harder water
While it is true that many parts of Europe have very hard water, many parts also have extremely soft water (like where I live). Detergent dosage recommendations take this into account and give doses for soft, medium and hard water.
I would suggest if anything you will need more than the recommended European dose for your water hardness, since American machines are so oversized. However, Considering they must be much more loosely loaded than a European machine, the load sizes (and therefore amount of detergent required) should work out about the same.
Other than that, yes it will work just fine in your machine.
Matt