Tazmanian Devil, you need to think more on what I wrote. This is not something which just came out of my mind but is the exact scientific explanation.
Yes, Moon does orbit the Sun, but that is not my point, it doesn't have an independent orbit like that of an asteroid or a planet. In fact it is part of gravitationally bound Earth + Moon system which together orbit the Sun. So there is a difference in having an independent orbit or orbiting by being part of a gravitationally bound system.
If a gravitationally bound system like Earth and Moon have to separate by a force external to this system (like Sun's gravitational force), it is not the intensity of Sun's force on any particular body - Moon or Earth that counts. So even if Sun pull's Moon twice as stronger than that of Earth on Moon, still the Moon does not need to separate from Earth.
It will separate only if the tidal forces of Sun on Earth-Moon system trying to pull away Moon from Earth are stronger than the gravitational force of Earth on Moon. In other words both Earth and Moon are accelerating towards the Sun, it is the difference in the acceleration, i.e., the net acceleration of separation of Moon from Earth due to Sun (in other words tidal force) that counts. This should be greater than the acceleration due to gravity that binds Moon to the Earth.
Tidal force of Sun separating Moon from Earth = Mass of Moon * ( accln due to gravity of Sun on Moon - accln due to gravity of Sun on Earth). This tidal force should be greater than the gravitational force that binds the Earth and Moon system.
Edited by 2gud, : No reason given.
Edited by 2gud, : Re-phrased certain sentences