Hello Animalrescuelady2451,
Thanks for posting your question about Mail on the Apple Watch. I'm happy to assist.
To clarify, what's the exact wording of the prompt you're seeing? Does it mention signing in to complete a purchase, or does it specifically mention Mail? If it's a purchase-related message, it's likely coming up because there's a pending update for an app that was installed using a different Apple ID. In that case, your best bet is to try updating each app one at a time to work out which one is causing the prompt: Install apps on your Apple Watch
If it's related to Mail only, there might be a few things to consider. What happens when you enter your password? Are you able to access the same account on your paired iPhone? How about if you open a web browser on your computer and sign in to webmail there?
Also, which watchOS and iOS versions are you using?
Take care.