Apparently, there are different versions at different sites on the internet, because the version I read said that the British invited the Americans to send a boat to the race, not that the Americans challenged the British to a race. I suppose it depends on whether it is an American or a Brit writing the history, eh?
Anyway, I think everyone agrees that the Cup was named after the boat that won it, not after a specific country. And probably a lot of people don't realize that -- I didn't know it until sometime within the last 10 years. I had always assumed that America must have started the event and that is why it was called that.