Referring to "the assertion made in the US Supreme Court's majority opinion"

Solution 1:

I'm not sure if this exactly answers your question, but I read a lot of legalese as part of my day job, and I've never seen a reference to any court's "claim." Plaintiffs and defendants make many "claims," but courts' opinions contain findings and conclusions. "Claims" has an aura of advocacy -- something the courts don't engage in (not officially, at least).

Also, depending on context, you might not need to say "majority." You only need to specify if you're quoting from a dissent or a concurring opinion, or if you're contrasting the majority opinion with a dissent or a concurrence.

If I were writing the comment you mention, I might say something like

The Court found that . . . .

I don't find anything grammatically wrong with your version, though.

Solution 2:

If you find that a possessive form is clumsy, as for instance because multiple possessives would end up being used in the same sentence, you can generally rephrase the possessive structure as a prepositional phrase or similar construct. Usually A's B can be alternatively phrased as B of A or perhaps as B belonging to A if the idea of ownership needs to be emphasized. So you might choose to transform one or both of the possessives in the Court's majority opinion's claim; if both possessive were to be eliminated, the result might be the claim in the majority opinion of the Court.