An interesting question indeed! I wouldn’t want to do your homework for you by giving specific examples from Justice Rehnquist’s opinion; however, here are some types of logical fallacies that I think you might find very useful as you think about this example.
“Using and Abusing Tradition”
Just because something has always been done one way or another is not in and of itself sufficient reason to continue doing it. A practice is not necessarily good, or necessarily bad, just because it has gone on for a long time.
“The Straw-Man Fallacy”
One of the best logical fallacies to be aware of as you go through life, the straw-man fallacy is used all the time in our political discourse and probably also in your arguments with your friend. This fallacy is characterized by misrepresenting your opponent's position, and then proceeding to knock down a lesser version of it, and declaring yours better. Philosophy professors will often instruct their students to present the best possible version of their opponents' arguments—this is to avoid using Straw Man arguments.
“The Democratic Fallacy”
Just because the majority of a population hold a certain view doesn’t mean it’s true. It might be interesting survey data, but provides no specific evidence one way or another.
“Tears as a Diversionary Tactic”
This fallacy involves obscuring the root issues or arguments by appealing to your audience’s emotional sensibilities. It’s very similar to the technique of pathos that you may learn about in a Rhetoric, English, or Public-speaking class. While a fantastic tactic to get people on your side, the use of emotion won’t have any bearing on the correctness or falsity of an argument.
These fallacies come from the book “Being Logical” by D.Q McInerny, which you might find highly useful for this assignment!