His power here is to set a direction and to nominate new appointees. He could write a bill to expand the bench and/or a constitutional amendment to require a code of ethics… Hell, he could even say “ok supreme Court, you say you can self-regulate… Publish your own code of conduct publicly or I’ll lead the charge in imposing one on you”
Presidents have a lot of soft power. He can write executive orders to demand the problem be evaluated, or he can use his platform to rally support… He can even go to Thomas privately and suggest he resign with dignity while he can, even try to bluff him off the bench
There’s a lot he could do - his hard power over the supreme Court is very limited, but soft power is how most everything works