Could Jupiter's gravity destabilize Earth's artificial satellites over a long timescale?
After all, it can destabilize Mercury's orbit, and it can also destabilize orbits in the asteroid belt.
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The Moon and, to a lesser degree, the Sun are the main perturbers, not Jupiter. The pertubation from the oblateness of the Earth is also significant. For the rate of change of right ascension of ascending node (RAAN) and argument of perigee for satellites above the geostationary orbit the Moon and Sun dominates and below geostationary orbit Earth's oblateness dominates.
Air drag for satellites in low orbits significantly affects the lifetime. As a typical example, the lifetime is 720 days for a circular orbit of 500 km (ballistic coeffient of 80 kg/m^3) . As the lifetime gets about 10 times longer for every 160 km it is about 2000 years for an orbit of 1000 km altitude. For geostationary orbits air drag becomes negligible, the lifetime is 10^(35000 km/160 km) * 2000 years = 10^219 * 2000 years (presuming this model holds).
 Space Mission Analysis and Design. Edited by James R. Wertz and Wiley J, Larsson. ISBN 0-7923-0971-5. Page. 126.
 Spacecraft Attitude Determination and Control. Edited by James R. Wertz. ISBN 90-277-1204-2. Page 64.
Most if not all of Earth's artificial satellites are unstable all on their own, requiring artificial maneuvering or maintenance not to fall back to Earth. So... whatever Jupiter would do seems negligible compared to their intrinsic instabilities, and definitely negligible compared to their/our capabilities for (self-)correction during their intended service life.