He only has proficiency if he caught the Missile with Deflect Missile and immediately throws it afterwards
As per the PHB page 78:
... If you reduce the damage to 0, you can catch the missile if it is small enough for you to hold in one hand and you have at least one hand free. If you catch a missile in this way, you can spend 1 ki point to make a ranged attack with the weapon or piece of ammunition you just caught, as part of the same reaction. You make this attack with proficiency, regardless of your weapon proficiencies, and the missile counts as a monk weapon for the attack.
This isn't weird at all considering you use:
- 1 ki point, a valuable resource for a Monk.
- A Reaction to catch (and be able to reduce the damage to 0)
So it's like you're catching the missile and using its momentum to throw it back in one swift action (or Reaction, excuse me). It's this skill that the Monk is proficient with.
For all other circumstances, the Monk is not automatically proficient with the Missile
This would fall on the Improvised Weapons rules in the PHB page 147:
... In many cases, an improvised weapon is similar to an actual weapon and can be treated as such. For example, a table leg is akin to a club. At your DM's option, a character proficient with a weapon can use a similar object as if it were that weapon and use his or her proficiency bonus.
So it's up to your DM.
You could argue that Arrows are akin to a Dart, which the Monk is proficient with but, looking at the Darts typically used in medieval times:
“Römischer Plumbata” by MittlererWeg is licensed under CC BY-SA 3.0 Unported
It does resemble an Arrow but with the most important distinction of being weighted for throwing (see that metal in the middle?). A normal Arrow just wouldn't be balanced enough to warrant throwing hence the need to spend a ki point to throw it after catching it ;)
Thanks, SevenSidedDie, for providing a legal image.