The Apple Watch is out in a handful of countries around the world, selling-out, earning great reviews and still generally bewildering people as to what it's really for.
One thing it's definitely not for is effectively browsing the web, as you can see from this video where jail-breaking fellow Comex tries to run something functional on a tiny 1.5 inch screen. It's a bit of a fail. And you'll really want to make the video fullscreen because its a bit borked.
It is definitely an iOS browser on a tiny screen, you can zoom and click-and-hold and... that's about it. There's no keyboard function so you can't actually search for anything but if you were browsing on your phone you might be able see some results on this screen as well - though why you'd want to is beyond me.
For right now, smartwatches really should just be an extension of certain parts of your phone - mostly notifications and some basic controls. Beyond that and you're getting into territory where it just makes a lot more sense to reach into your pocket for your handset.