If Spock came with a super machine capable of moving planets, and placed Earth so close to Jupiter as the Earth's moon, the gravity of Jupiter would make us fall into it?
That's no good idea. Earth wouldn't necessarily fall into Jupiter in the short run, provided it orbits Jupiter fast enough (within about 1.7 days), and on a circular orbit, but we would risk to collide with Io, destroy it by tidal forces, or change its orbit heavily. The other Galilean moons would get out of sync and change their orbits over time.
Tides would be severe on Earth, not just limited to oceans, but also for "solid" ground, as long as Earth isn't tidally locked. This would result in severe earth quakes and volcanism.
Our days would be dim due to the distance to the sun. After tidal locking of Earth and ejection/destruction of Io the tides as heat source would be lost, oceans would freeze, temperature would fall to about -160°C mean temperature. During the polar night oxygen would probably condense from the atmosphere and form lakes, may be even nitrogen. By this atmospheric pressure would drop.
Since Spock is smart enough to know these consequences in advance, he wouldn't do it.