These instructions apply to all versions of the Apple Watch and watchOS.
The Apple Watch comes with between up to 32 GB of storage space, depending on the model. You aren't likely to fill all of that up with texts alone, but Messages are one place you can clear some room if the need arises. Considering how tedious the process is, you should consider deleting text messages as one among several ways to free up some data on your device.
It would make more sense and take less time to delete all messages from the Apple Watch at once. Unfortunately, a clean sweep of Messages isn't an option. You can, however, delete conversations one at a time. Here's how.