Although Apple introduced an over-the-air (OTA) mechanism in watchOS beta 1, software updates still require an iPhone. But it is still a beta, after all (via MacRumors).
watchOS 6 B1
On watchOS 6 beta 1 you’ll find the OTA mechanism under Settings > General > Software Update. But you’ll have to review and agree to the terms and conditions on your iPhone.
The biggest change to having the Apple Watch be an independent device is, of course, its own App Store which will arrive with watchOS 6.