I have tried to compare between the Earth/Moon to Sun/Moon gravity force as follow:

Earth Mass = M⊕ = (5.9722±0.0006)×1024 kg

Moon Mass 7.342×1022 kg (0.012300 of Earth's) = 0.0123 M⊕

Sun Mass = M☉ = 1.98855×1030 kg (333,000 × Earth) = 333,000 M⊕

Radius (moon-earth) = 384399 km = 0.00257 AU = 2.57 x 10 ^ -3 AU

Radius (earth-sun) = 149,597,887 km = 1 AU

As the Moon is located close to Earth, than we can assume that this value also represents the Radius (moon-sun)

Therefore,

Radius (moon-sun) = Radius (earth-sun) = 149,597,887 km = 1 AU

Gravity Force:

F = G x M1 x M2 / R ^ 2

F (Earth Moon) = G x M(moon) x M⊕ / Radius (moon-earth) ^ 2 =

G x 0.0123 M⊕ x M⊕ / (0.00257 AU) ^ 2 = G x M⊕ ^ 2 x 0.0123 / (6.6 10 ^-6 x AU ^ 2) =

G x M⊕ ^ 2 x AU ^ -2 x 1,863.63

F (Sun Moon) = G x M⊕ x M☉ / Radius (moon-sun) ^ 2 =

G x M⊕ x 333,000 x M⊕ / (1 AU) ^ 2 =

G x M⊕ ^ 2 x AU ^ -2 x 333,000

Therefore, with regards to the Moon;

The direct Gravity force from the sun is stronger then the gravity force from the Earth:

F (Sun Moon) / F (Earth Moon) = 330,000 / 1,863.63 = 177

So, why our dear moon had decided to orbit the Earth, while it can get a 177 stronger gravity force directly from the Sun?