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The Moon draws rotational momentum from the Earth; if that lasts long enough, the Moon will cease to be a satellite of the Earth. If it isn't enough, the Earth and Moon will be locked in a face-to-face resonance. The loss of the Moon will not cause instability, it will if anything lead to greater stability -- after the Moon wanders away far enough to wander off. The transition will be, uh, a little rocky, and that will last a while.

"The Moon has a stabilizing effect on Earth's obliquity [tilt angle]. Frequency map analysis suggests that, in the absence of the Moon, the obliquity can change rapidly due to orbital resonances and chaotic behavior of the Solar System, reaching as high as 90 degrees in as little as a few million years (also see Orbit of the Moon).[24][25]

However, more recent numerical simulations[26] suggest that even in the absence of the Moon, Earth's obliquity could be considerably more stable; varying only by about 20-25 degrees [23.5 + 20-25 deg].

The Moon's stabilizing effect will continue for less than 2 billion years.

If the Moon continues to recede from Earth due to tidal acceleration, resonances may occur which will cause large oscillations of the obliquity.[27]"

[27] Ward, W.R. (1982). "Comments on the Long-Term Stability of the Earth's Obliquity". Icarus 50: 444-448. Bibcode:1982Icar...50..444W. doi:10.1016/0019-1035(82)90134-8.

https://en.wikipedia.org/wiki/Axial_tilt
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"The pull of the moon is also slowing the Earth's rotation, an effect known as tidal braking, which increases the length of our day by 2.3 milliseconds per century.

The energy that Earth loses is picked up by the moon, increasing its distance from the Earth, which means the moon gets farther away by 3.8 centimeters annually.

The moon's gravitational pull may have been key to making Earth a livable planet by moderating the degree of wobble in Earth's axial tilt, which led to a relatively stable climate over billions of years where life could flourish."

http://www.space.com/55-earths-moon-formation-composition-and-orbit.html/

Stabilization of the Earth's obliquity by the Moon

J. Laskar, F. Joutel & P. Robutel

Astronomie et Systemes Dynamiques, Bureau des Longitudes, 77 Avenue Denfert-Rochereau, F75014 Paris, France

ACCORDING to Milankovitch theory [1,2], the ice ages are related to variations of insolation in northern latitudes resulting from changes in the Earth's orbital and orientation parameters (precession, eccentricity and obliquity).

Here we investigate the stability of the Earth's orientation for all possible values of the initial obliquity, by integrating the equations of precession of the Earth. We find a large chaotic zone which extends from 60 deg to 90 deg in obliquity.

In its present state, the Earth avoids this chaotic zone and its obliquity is essentially stable, exhibiting only small variations of +/- 1.3 deg around the mean value of 23.3 deg.

But if the Moon were not present, the torque exerted on the Earth would be smaller, and the chaotic zone would then extend from nearly 0 deg up to about 85 deg.

Thus, had the planet not acquired the Moon, large variations in obliquity resulting from its chaotic behaviour might have driven dramatic changes in climate.

In this sense one might consider the Moon to act as a potential climate regulator for the Earth.

http://www.nature.com/nature/journal/v361/n6413/abs/361615a0.html