Does updating iOS force you to update watchOS?
No, it won't force you to update your Apple Watch. You can use an Apple Watch with older versions of watchOS while using iOS 10 on your iPhone.

You won't be able to use any features that are exclusive to iOS 10 or higher watchOS versions, though. Think standalone watchOS apps (watchOS 2.x and higher), iMessage animations (watchOS 3.x and higher), new watch faces and so on.