No. Because they specifically said this is not the case.
The President enjoys no immunity for his unofficial acts, and not everything the President does is official. The President is not above the law. But under our system of separated powers, the President may not be prosecuted for exercising his core constitutional powers, and he is entitled to at least presumptive immunity from prosecution for his official acts.
They're essentially protecting a president from flagrant lawsuits that could be brought for unfounded accusations. The constitution outlines a handful of constitutional duties (such as pardoning) which are by definition the law not prosecutable. There's a presumption of immunity for their official acts. Anything they do outside of official acts is not immune.
Nothing has really changed. It's only made it more clear how difficult the process is to indict a president. The Fourth section of Article II still exists.
The President, Vice President and all civil Officers of the United States, shall be removed from Office on Impeachment for, and Conviction of, Treason, Bribery, or other high Crimes and Misdemeanors.
So, let's say, not for the first time ever, a president orders an assassination and congress wants to hold them accountable for this action. It will need to be determined if this act was part of their official duties. The issue SCOTUS has presented is that it's very, very difficult for congress to obtain the motivation for such an act. Such a case would be dependent on the specific circumstances. I mean, if the president orders the assassination of a foreign leader, no one's going to, nor have the ever, question that. If they order the assassination of a congressional leader, don't imagine they're going to get away with that.
