“How does an impeachment by the House stop any Supreme Court nomination?”
Perhaps they’d use the argument that they trotted out last time that a President who is impeached should not be allowed to appoint Judges/Justices until/unless he is acquitted, hoping to peel off a squishy Republican Senator or two—they don’t need that many.
That could apply only in the event of the US SZenate voting to convict. Zero chance of that.
Be that as it may, the president still retains the right to appoint Supreme Justices.