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.
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https://twitter.com/horwitz/status/1140700946500116481
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.
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