Context

The paladin in my game frequently uses his backhand to "correct" the ranger/assassin/warlock. The table enjoys these social interactions, as part of their bromance. It mostly happens out of combat, but we do still roll damage for the slaps. Since the paladin is not the smartest of the bunch this could potentially backfire if he hurts his ally too much (or at the wrong time), especially since the ranger is now vulnerable to radiant damage (due to homebrew events).

According to How much damage does a punch deal?, by default:

An unarmed strike deals 1 + Strength modifier damage.

Improved Divine Smite (IDS)

By 11th level, whenever you hit a creature with a melee weapon, the creature takes an extra 1d8 radiant damage.

Confusion

I read somewhere that an unarmed strike counts as a simple melee weapon and therefore allows you to add proficiency bonus to attack rolls if the (N)PC is proficient in simple melee weapons. But unfortunately I can't find the source for that statement. If this is true, it implies the IDS would trigger when hitting with unarmed strikes.

Research, with sources:

• 
  

  Is an Unarmed Strike considered a Weapon attack?

  
  
  

  
  

  Instead of using a weapon to make a melee weapon attack, you can use an unarmed strike: a punch, kick, head-butt, or similar forceful blow (none of which count as weapons).

  
  

  
  


• 
  

  For a basic unarmed strike, is only the proficiency bonus included in the attack roll, or is the Strength modifier also added?

  
  
  

  
  

  For example, an unarmed strike counts as a melee weapon attack, even though the attacker’s body isn’t considered a weapon.

  
  

  
  

Question

Both accepted answers suggest that the slapping hand itself would not count as a weapon, and would therefore not trigger IDS, as the back of a hand is not a melee weapon. Although these answers hint towards my conclusion, they don't give a clear yes or no to my question.

Does Improved Divine Smite trigger when a paladin makes an unarmed strike?

In other words, is my current conclusion correct? Or am I misinterpreting the wording of the paladin feature?

Improved Divine Smite will not trigger on an unarmed strike.

(luckily for the other PC)

Your conclusions are correct. As the rules state, an unarmed strike is not considered a weapon of any kind. According to the PHB rules on melee attacks (post-2016 errata):

Instead of using a weapon to make a melee weapon attack, you can use an unarmed strike: a punch, kick, head-butt, or similar forceful blow (none of which count as weapons). (PHB 195)

^{(That same errata also removed unarmed strikes from the weapons table in the PHB (p 149).)}

Improved Divine Smite requires a weapon:

whenever you hit a creature with a melee weapon

Since Improved Divine Smite requires a weapon, it will not work with unarmed strikes.

Note that normal Divine Smite does still work with unarmed strikes.

Normal Divine Smite is written differently in a slight but significant way. It says:

[...] when you hit a creature with a melee weapon attack [...]

Hitting a creature with a melee weapon explicitly requires a melee weapon. However, there are things that can do melee weapon attacks without being considered weapons. Unarmed strikes are one of those things.

As the Sage Advice Compendium clarifies:

an unarmed strike counts as a melee weapon attack, even though the attacker’s body isn’t considered a weapon.

So you would be able to apply Divine Smite to an unarmed strike, but not Improved Divine Smite.

You always add your proficiency bonus to your unarmed strikes

The rules for melee attacks state:

On a hit, an unarmed strike deals bludgeoning damage equal to 1 + your Strength modifier. You are proficient with your unarmed strikes. (PHB 195)

This line was also added in the 2016 errata and may not appear in your printed PHB.

So, no matter what proficiencies you have you always add your proficiency bonus to an unarmed strike attack roll. This does not affect any of the rulings on (Improved) Divine Strike though.

Divite Smite works with an unarmed strike but Improved Divine Smite doesn't.

Let's break down the reasons:

Divine Smite

As the author of the blog here writes:

What does this mean? Unarmed strikes are melee weapon attacks but
not weapons. There’s been a bit of confusion as to whether this mean
stunning strike works with unarmed strikes – it certainly does, the
strike is still a melee weapon attack! (Stunning Strike says “When you
hit another creature with a melee weapon attack”. You use a melee
weapon attack to make unarmed strikes, so it still applies. The same
applies to Divine Smite – you may use that with an unarmed strike.

This is supported by the 2019 SA Compendium.

Improved Divine Smite

The wording of Improved Divine Smite states:

By 11th level, whenever you hit a creature with a melee weapon, the creature takes an extra 1d8 radiant damage.

As we've already seen:

Unarmed strikes are melee weapon attacks but not weapons.

Hence you won't get the 1d8 radiant damage on every unarmed strike.