Can copyright prevent claims of self-plagiarism? So, I had somewhat of a thought experiment idea.

I'll be honest: I define plagiarism as misrepresenting who wrote (or provided the ideas of) parts of your papers. I don't believe you can self-plagiarize. If the goal of the assignment is to teach you something, and you already learned it in the past (by writing a paper that fits), I see no point in the busy work of making the current paper different enough just for the sake of it.

I realized that in the US, authors are granted a copyright to their creative works by default. (Even if few of them realize this or ever claim copyright to their works.)

Could a student use their copyright of their works to prevent (or at least seriously restrict and complicate) faculty's checking of self-plagiarism in the future? Couldn't the student just not grant the right to copy their works? If the teacher/professor hands graded work back, then they would have no copy left, and if they don't, they would have to share that one original copy with everyone who wants to check a student for self-plagiarism.

Personally, I have never (in 3+ years) had a case where I could reuse (parts of) a paper. I haven't yet had instructor specify that assignments had to be written for that assignment only. If an instructor did specify that, I would follow their rules. I would, however, try to convince them to change their mind, and this thought experiment might be part of that discussion.
First off:


  I don't believe you can self-plagiarize.


I'm afraid that it doesn't matter what you believe about the word or its definition. It is not the reuse itself that is considered here, but it matters that most other academics believe that unattributed reuse of one's own work is ethically incorrect. If you choose to do things that most others label "self-plagiarism," then sooner or later it will not go well for you.

Now, as for your idea about copyright: potentially you could in fact make yourself a pain for the faculty members who deal with you.  You don't even have to be right or sane about your argument, you just have to threaten to make a big legal stink.  You will then greatly complicate your professor's lives, but one way or another you will not prevail: most likely either you will lose the legal case or the university may simply decline to continue instructing you.