In due time, would both of them collide if they were affected solely by the gravitational force of each other?
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No. In fact, due to tidal acceleration some of the Earth's rotational angular momentum is being transferred to the Moon's orbit, and so the Moon is gradually moving further away from the Earth (and the Earth's rotation is slowing down). This is predicted to continue for billions of years until the two are tidally locked and Earth's rotation has slowed to match the duration of the Moon's orbit. However, by that time the Sun will have expanded to a red giant, so predicting what will happen past then is difficult.
Eric Smith
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