From Clinton v. Jones 520 U.S. 681 (1997):
Deferral of this litigation until petitioner's Presidency ends is not constitutionally required.
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The separation-of-powers doctrine does not require federal courts to stay all private actions against the President until he leaves office.
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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.
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The President's absolute immunity extends to all acts within the "outer perimeter" of his duties of office.