I was just following your lead on that. I wasn't making that claim myself.
If they didn't appoint themselves to that role, then who did?
It's not a matter of appointment. This third branch of government simply said that they, too, have a place in determining what is constitutional law.
The legislature, when writing a bill, is cognizant of constitutional considerations, as is the President when he signs it. Both branches have a sworn duty to uphold the Constitution, and both branches may contest the constitutionality of a proposed piece of legislation.
The USSC only sees the legislation after it becomes law and is contested. One can say that they are the "final" arbiter, but that is only because they are the last signatory. Marbury v Madison established the USSC as one of the three branches making a constitutional determination.