There is no hard law (statutory, constitutional or SCOTUS holding) regarding "conflict of interest" for Supreme Court justices. Justices have typically recused themselves in case of financial interest, for example owning shares of a plaintiff / defendant company. Also, if a justice was involved in lower-level proceedings (e.g. circuit court appeal), they will step aside. If a justice is a defendant in a case, they (tend to) recuse themselves – the one relevant case that I know is Jaffe v. Roberts et al. where somebody sued all of the justices except Gorsuch, who was the new kid on the block, so the lower court judgment was affirmed by default. Justices have exercised the option of recusing themselves because of other connections to a party, such as a close friend being a party, a party being one's son's educational institution, a party having testified against th justice at a confirmation hearing. There is a case (Jewell Ridge Coal Corp. v. United Mine Workers of America) where Hugo Black participated, but a former law partner of Black's argued for the winning side. This is one of the controversial cases of non-recusal.
The ethical guidelines are not bright except in cases of financial interest, prior involvement, or being an actual party. No SCOTUS justice has recused themselves because the administration which nominated them is also being sued, and it would not be expected that a justice nominated by a president would recuse themselves from hearing. In Clinton v. Jones, Ruth Bader Ginsburg and Stephen Breyer who were Clinton appointees participated in the decision, and voted against Clinton. In US v. Nixon, Rehnquist recused himself because of association with Watergate defendants, and not because Nixon nominated him. Lewis Powell, nominated by Nixon, did not recuse himself and voted against Nixon, idem Harry Blackmun.
