I was able to download apps from the App Store when I paired my watch at first. However, I eventually ran into exactly the same problem as the author, but I figured out a way to make it work again!
TLDR: Retry it after making sure you have Apple Wallet app installed on your Apple Watch.
(I found that re-pairing the watch also works because it forces the watch to come with Apple Wallet preinstalled after reset. But re-pairing is unnecessary if all you need to do is to re-download the Apple Wallet app.)
My issue was that I deleted the Apple Wallet app when I was getting rid of some apps that I don't use from the grid view. I downloaded the Apple Wallet app from the app store again (which does not require you to double click the side button since it is an app you've already had before, hence the cloud icon with an arrow instead of the "GET" button), and I attempted to get the app I wanted. Double click responded; It worked!
My guess is that the App Store knew it needed to go through Apple Pay to mark it as "purchased" (even if it is free), but it couldn't find the Apple Wallet app to do that, so it just didn't react properly. This behavior is confusing. I think apple should've made it so that it shows the user a clear error message saying the Apple Wallet app is missing instead of making it look like the double click is just unresponsive.
Also note:
- You need a passcode set up for the watch. (You can turn it off later and turn it back on whenever as you need to.)
- Check if you are double clicking the side button with the right speed. (You can get a feel for the rhythm and even adjust the speed by navigating to "Accessibility" > "Side Button Click Speed" (under "MOTOR").)
- If you have "Confirm with AssistiveTouch" set to on, turn it off
(in "Accessibility" > "AssistiveTouch" (under "MOTOR")).
- You need to be connected to the internet. (Try using Siri or something to make sure it indeed is connected.)
- You need to be signed into your Apple account. (Maybe go to "Settings" > [your name] > click around buttons within it to see if you can force it to make you log in or something. Maybe also remove App Store from the App Switcher to relaunch it too.)
- If it still doesn't work, try again after turning off and on the watch for a good measure.
If all of that didn't solve the issue, I don't know what other non-hardware fix you can do to fix it other than resetting and re-pairing the watch.
