Technically they could, but it would mean creating and managing a new license fee system.
Right now there's no enforcement of the license fee, just the knowledge that, to watch live broadcasts (either on TV or online) you must have a license, and if a TV license inspector catches you, you'll be in trouble.
With the iPlayer service, if you watch any of the live-streaming channels and you don't have a TV License you will eventually get a letter informing you that you need to pay the license fee. Fail to obtain the TV License after that point and you are subject to the normal fines.
I think the BBC must have some form of tie-in with the ISPs as I know a number of people that have had this happen to them.
I'm regularly tunnelling all my traffic through a UK VPS (not for geolocation regions, I'm in the UK myself anyway), and never had any issues watching iPlayer.
And if I watched to avoid it, I could always use something like www.tvcatchup.com to watch the live content, and just go onto the iPlayer for VOD content (for which you don't need a license).
I wonder - traditionally a TV license is per household, not per person, meaning that if I want to watch on a TV in a friend's house, they need to have a license, not me. With internet access, how does that work. For example, say they were able to prove I had been watching iPlayer live streams in my house, what if it was a friend watching on their laptop, and they own a license? What if you're watching from a public WiFi or a 3G connection on a train? The "license for the house" doesn't quite translate.
They'd still need to differentiate between shows that they could show to a US viewer and ones which they couldn't (because they didn't have the rights to international distribution - for instance most sport, any drama they've bought in and so on).