Why doesn't earth's moon spin on its axis?
The Earth's Moon does spin on its axis.
The Moon is what is known as tidally locked to the Earth. This means that its rotation period is the same as the time it takes the Moon to orbit the Earth.
The Earth's gravity induced tidal forces in the Moon which slowed it down until it was locked. All Moons become tidally locked to their parent planet in time.
In fact the Moon's gravity is slowing the rotation of the Earth by tidal forces. The day is getting longer and eventually the Earth will be tidally locked to the Moon. This will mean that the day will be a month long and only one half of the Earth will be able to see the Moon.