I am not a copyright lawyer but I have researched this question quite a bit. The simplest thing I can tell you, at least as far as US law is concerned, is that the question of copyright boils down to what you can prove in a court of law should someone challenge you or you should challenge someone else over the question of intellectual property rights.

Basically, as soon as you design a thing, you have the copyright. The question is can you prove it if you have to?

Registering a design with the US Copyright office provides a record of your design that can help you do that. In terms of tartan, having it registered with the National Register, or recorded with the STA, also provides a public record of your design. Posting an image of it here on the forum also creates a record. Putting it in a sealed envelope and mailing it to yourself (so that it has the post date on it) also can establish when you created the work.

All of the above is just about establishing a way to prove that you did indeed design the thing when you said you did, in case that ever gets challenged.

And, to answer your more specific question, no, you don't have to have the tartan woven into cloth in order to establish copyright of your design. You are, asked, however, to have it woven in order to send a sample in to the National Archives if you are having the tartan registered, and it would be a good idea to also send a woven sample to the STA for their extensive tartan archives, as well. But that's quite different than establishing copyright.