Yes.. Always gravity exerted by an object that has larger mass is more.
The Earth pulls on the Moon, and the Moon pulls on the Earth. The Sun pulls on the Moon, and the Moon pulls on the Sun. Mars pulls on the Moon (ever so slightly) and the Moon pulls on Mars. Your body pulls on the Moon (hardly measurable, but it does) and the Moon pulls on your body. The Moon and Sun and Earth's gravity pull (and push) Earth's oceans, and the ocean "tides" are the results.
No, the Earth's gravity pulls the moon in towards Earth.
Gravity
sun's gravity pulls earth and earth pulls on the moon.
The Earth's gravity pulls the Moon towards it and its center.
gravity
The moon's gravity is one-sixth that of the Earth's;
The moon has one-sixth of the Earths gravity.
The moon's gravity creates high tides and low tides. The moon's gravity pulls the water up and down as it orbits the Earth.
The moon's gravity pulls on the oceans and causes tices.
The gravitational pull of the earth. Gravity from the Earth pulls the moon towards it.
The moons gravity 'pulls' the earths water creating a 'tide.'