You can’t use Samsung Galaxy Gear to browse the internet, because it has no separate data connection option. It is used as a partner device with Samsung Galaxy devices for making calls, text and for various other tasks. But if you want full internet access on it or want to install apps from Play Store or want to add live wallpaper you can do all that by installing custom ROM.
Galaxy Gear comes with an 800 MHz CPU, 512MB RAM, 4GB internal storage, speaker, microphone, Bluetooth, etc. so you can do everything our phone can do. We already knew that how to install android apps on Galaxy Gear, so by installing custom ROM you could use your Gear for small browsing purposes as well via Bluetooth tethering.
The custom ROM called Null for Galaxy Gear offers several features including Play Store access, full internet access via Bluetooth tethering, wallpaper support, native apk installation, weather widget, multiple devices Bluetooth pairing and more.
However, keep in mind that, installing custom ROM involves dangerous process, it may brick your device, void manufacture warranty, so do it at your own risk. You can get more information, download and installation guide at XDA forum.