Euro 2016 has an advantage being on ESPN, but there are arguably more in the us with family connections to western hemisphere nations than European nations. Also, the US has made (at least) a semi-final run at the Copa. Which tourney is doing "better?"
It depends on the game. If it's England, Germany, France, etc that might have a lot of ex-nationals or descendants, it's easily Euro. But, if it's Mexico or U.S., it's probably Copa. The one thing is, they haven't tended to overlap. You could watch games all day. Starting with Euro and finishing at night with Copa. The better time slot is with Copa America. So, they get an edge. I doubt there's 'soccer snobs' out there that will watch one but not the other (unless they don't get FS1 if that's the broadcaster).