Yeah, it's cheesy, and played out, and Hollywood. And it just doesn't sound like a name. I can't imagine calling out "River!" without sounding like somebody hollering "timber" as they cut a tree down. It just sounds silly, like a flood warning.
Also, do you mind me asking: does the baby's last name have an er ending? I ask because I've recently seen a couple people with er endings to their first and last names. Say Taylor Parker and Heather Keller. I mean, River Waters is obviously a really bad idea, but River Hunter or Rivers Connors would also sound silly. I call that style of combo Foghorn Leghorn names.
Before you criticize a man, walk a mile in his shoes. That way, when you criticize him, you're a mile away and you have his shoes!
Steve Martin