He did, during his swearing in.
I found it very amusing that the Democrats accused Kavanaugh of being partisan. Of course he's partisan. So is Ginsburg. So is Kagan. So, as we can see, is Sotomayor. There are Republican justices, there are Democrat justices, and anyone who thinks the Supreme Court is non-partisan is about 40 years behind the times.
You are right as far as you go, however, the big difference is that those Justices on the right apply the Constitution as it is written while those on the Left apply the Constitution as they wish it were written.