I'm considering a multi-class of Monk and Ranger, and have been trying to decide which Fighting Style I'd like to pick.

Two-Weapon Fighting and Defense are useless for a Monk, as they won't be wearing armour, and using multiple weapons stops unarmed strikes.

Dueling could work, if the character uses a 1-handed monk weapon, giving a +2 to damage with it (I believe).

As for Archery, I was wondering if catching and throwing something with Deflect Missiles would get the +2 to attack. The reason I believe it would is that it counts as a Monk Weapon for the attack.

Does this logic look right?

No, it doesn't

The Archery fighting style says:

You gain a +2 bonus to attack rolls you make with ranged weapons.

Ranged Weapons in this case refers to a weapon type, so you won't get the bonus. If this instead said, "Ranged Weapon Attacks", you'd get the bonus, but as is, you're out of luck.

This is the same reason why the Archery style doesn't work with most* thrown weapons -- they're melee weapons with the thrown property!

*Darts and Nets being the exceptions, because they are Ranged Weapons.

Unfortunately, No

The Archery fighting style requires you to make the attack with a ranged weapon:

You gain a +2 bonus to attack rolls you make with
ranged weapons.

And as clarified in the Sage Advice Compendium, a thrown melee weapon does not qualify:

Does the Archery fighting style work with a melee weapon that you throw? No, the Archery feature benefits ranged weapons. A melee weapon, such as a dagger or handaxe, is still a melee weapon when you make a ranged attack with it.

So, while the thrown object is a Monk Weapon, it would still be a thrown weapon (arrows as an improvised weapon technically), unless it was a ranged weapon already such as a dart or net.