From Clinton v. Jones 520 U.S. 681 (1997):
Deferral of this litigation until petitioner's Presidency ends is not constitutionally required.
The separation-of-powers doctrine does not require federal courts to stay all private actions against the President until he leaves office.
Nixon v Fitzgerald provides no support for an immunity for unofficial conduct.
Regarding official acts, the President is immune. Nixon v. Fitzgerald 457 U.S. 731 (1982):
Petitioner, as a former President of the United States, is entitled to absolute immunity from damages liability predicated on his official acts.
The President's absolute immunity extends to all acts within the "outer perimeter" of his duties of office.