Recently, Google mentioned in a blog post that an Android TV app for Google Stadia would be making its way to the new Chromecast, select Android TV OS devices, and unsupported Android TV OS devices through an experimental beta program.
It looks like yesterday, some users were reporting that you are now able to download the Stadia App for your Android TV devices, but it is still too early to play games on it.
Attempting to launch the application leads to a "Coming Soon" message.
With that being said, in the initial announcement from Google, it was mentioned that the Android TV OS app would go live on June 23rd. At the time of writing this article, that will be tomorrow.
Some users have stated they found a way to get around the "Coming Soon" screen, but it requires that you screencast from a mobile device. If you are unable to wait until tomorrow, you can check out these steps posted by 9to5Google - https://9to5google.com/2021/06/21/how-to-play-stadia-games-early-android-tv-google-chromecast/.