perigee
No, Earth's Moon it is the closest - since Mercury and Venus have no moons.
Venus
That would be Venus.
What fraction of earth's diameter would be closest to the moons diameter
Mercury is the planet that is one quarter the size of Earth and has no moons. It is the smallest planet in our solar system and is the closest planet to the Sun.
Our only moon, which is called `The Moon`. Our moons average distance from the Earth is 384,399 km (238,854 miles).
Mercury is closest to the Sun and it is not close to any moons
The moons distance from Earth varies during its orbit, but is around 363,100 km or 225,620 miles at its closest (Perigee). Although this seems quite far, its only a small distance when compared to the Earth sun distance, which is 147,098,290km or 91,402,600 miles at its closest. So the moons closest distance to the sun will be around 146,735,200km.
The moons average distance from earth is 384,400 kilometres (238,855 mi). The closest the moon gets is 356,700 kilometres (221,600 mi) and the furthest is 406,300 kilometres (252,500 mi).
The moons distance from earth varies throughout is orbit, ranging from approximately 357,000 kilometres (222,000 mi) and 406,000 km (252,000 mi). A supermoon is where this closest approach coincides with a full moon. So a supermoons distance from earth is 357,000 kilometres or 222,000 miles.
Charon.
There are no moons around Venus or Mercury. So the closest moon to Venus is the Moon orbiting the Earth, which is not really any closer than Earth is. (Venus is always at least 100 times farther away from Earth than the distance of the Moon.)