The executive nominates judges, but they hold tenure for life and after confirmation operate with complete independence as a separate branch of government with their own budget etc. The Supreme Court is also irrelevant to this because they do not initiate investigations, they do not oversee prosecutors, and they do not administer the justice system. They only hear cases about Constitutional issues - not criminal issues, so unless there's a constitutional dispute a case has no chance of reaching them.
Judges in general do not oversee or initiate prosecutions, U.S. attorneys do. They're also appointed by the executive, but they operate independently and while, subject to Congressional oversight, he can fire some of them (but not the permanent civil servants who are the bulk of the staff), he cannot directly order them to do anything.
In any case, an investigation of this complexity almost certainly started under Obama, just because they tend to take many years to reach the stage where charges are filed.
Trump frequently claims to have powers that he does not have. He has said many times that he is going to do something only to find that he cannot actually do it, and so it hasn't been done.