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So a positive/negative charge can attract a neutral object by causing the negative/positive charges to move closer to the source.

However, if you have a neutral object and another neutral object placed next to each other, can't, for example, all the electrons on the left neutral object move to one side, and similarly with the other neutral object so that the two objects become charged?

For example +- -+ ---> +- +-

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    $\begingroup$ See Hydrogen bond (water molecules are neutral, yet there is a non-zero electrostatic attraction between molecules). $\endgroup$
    – AccidentalFourierTransform
    Commented Jan 16, 2016 at 19:20

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Yes, this happens and it's called London Dispersion force.

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If the two neutral objects are conducting spheres, for example, the charges spread out over the surface because like-charges repel. The negative/positive charges don't want to be clumped together in one place if they can help it. So a neutral sphere will not polarize unless a net charge pushes/pulls the negatives to one side and the positives to another. But a neutral object does not have a net charge, so it cannot separate the charges in another neutral object.

If you have something like a water molecule, however, the relative positions of the charges are fixed by the shape of the molecule. And so two water molecules can be neutral but polarized, and so attract or repel each other. But the water molecules are already polarized by their shape; they don't polarize each other.

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