This particular smartwatch update was not the most intuitive of the bunch. Typically, I receive Samsung watch updates through the Wear app or the Google Play Store. The Google Assistant installs through the latter method, but you have to search for it on the watch itself.
On your Galaxy Watch 4, swipe up on the main watch face to get into Wear OS’s app drawer. From there, select the Google Play Store app. Once it’s launched, tap on the Search icon at the very top of the window, and type in “Google Assistant” on the watch’s display. Search for that, then tap on Google Assistant when it appears, and select the option to Install (in some cases, you might see an Update button instead).
After it installs, you can fire up the Google Assistant from the same application drawer to get it configured or tap on Open from the Play Store menu screen on the watch. You’ll be prompted to open up your phone to finish the installation.