The new Apple Watch functionality comes as a result of the version 4.1 update to the app, and while you are able to make purchases, you won't have access to the entire Apple Store catalog. As it stands, you can only purchase accessories from the Apple Watch, such as cases, headphones, and more. It'd be nice to have complete access to all of the offerings, but buying a $40 case from your watch makes a lot more sense than purchasing a $1000 computer.
In addition to the ability to purchase items, the update to the Apple Store app also includes support for rich notifications on iOS 10.
This update brings the total size of the Apple Store app up to 70.9MB, and you'll need to ensure that you're running watchOS 3 in order for the new Apple Watch features to work. iOS 9 is the bare minimum that you need to download the app, but rich notification support is only available if you're running iOS 10.