A company with European trademark disputes my U.S. based website for infringement. Do they have an argument?

You are obviously aware of some of the legal issues involved here. You are correct to seize on defenses that undermine any such claim. You identify three such defenses: territorial scope, absence likelihood of confusion, and generic terms (the legal term is "descriptive" not generic.) With regard to their being in Europe, in the absence of a US TM registration they may not be able to affect your use of the brand name in your business, with one major exception, your domain name. They may attempt to block and divest you of your domain name through international litigation, or more likely through the Uniform Dispute Resolution Policy. The latter is an arbitration proceeding that everyone with a domain name contractually submits to, when they register. Since, the UDRP is relatively cheap (esp. in terms of international enforcement), it is something you should be concerned with. The crux of that proceeding is that the claimant asserts superior trademark rights to the tradename embodied in the domain against the other party. Hence, US and other trademark doctrines and defenses will need to be assessed to protect against that possibility.

As for the terms in the brand name being generic that might be asserted as a defense. In addition, absence of confusion, stemming from the difference in services is another defense that you would assert. However, the most potent defense is senior use, meaning that you used the domain prior to them.

There may be more defenses that you should consult with an attorney to assess. And if you are intent on protecting your domain and brand name you should certainly have that attorney send a response in writing that affirms your commitment to those to deter them from further action.

I hope this is helpful.

Disclaimer: This answer is for informational purposes only and does not constitute general or specific legal advice, nor create an attorney client relationship.