Is a melee spell attack also a melee attack?

Question

The absorb elements spell says:

The spell captures some of the incoming energy, lessening its effect on you and storing it for your next melee attack. You have resistance to the triggering damage type until the start of your next turn. Also, the fist time you hit with a melee attack on your next turn, the target takes an extra 1d6 damage of the triggering type, and the spell ends.

The shocking grasp cantrip says:

Lightning springs from your hand to deliver a shock to a creature you try to touch. Make a melee spell attack against the target. You have advantage on the attack roll if the target is wearing armor made of metal. On a hit, the target takes 1d8 lightning damage, and it can’t take reactions until the start of its next turn.

If I absorb fire damage with absorb elements, would I be able to add an extra 1d6 fire damage to my next shocking grasp attack?

Answer 1

Yes, melee spell attacks are melee attacks

The rules define melee attacks to include both melee weapon attacks and melee spell attacks (PHB 195):

Used in hand-to-hand combat, a melee attack allows you to attack a foe within your reach. A melee attack typically uses a handheld weapon such as a sword, a warhammer, or an axe. A typical monster makes a melee attack when it strikes with its claws, horns, teeth, tentacles, or other body part. A few spells also involve making a melee attack.

Also, I found a relevant Q&A Twitter post from Jeremy Crawford (the Lead Designer and Managing Editor for D&D 5e) that sheds some light on this. Crawford's tweets are no longer official rulings as of the 2019 Sage Advice Compendium, but his response quoted below is still useful guidance that matches what is clearly stated in the official rules.

Question

Does a "melee spell attack" count as a "melee attack" for Touch of Death?

Answer

A melee spell attack is, indeed, a melee attack and can qualify for the Death cleric's Touch of Death feature.

This would naturally extend to other effects that alter or activate based on melee attacks.

Answer 2

Yes, melee spell attacks are melee attacks

Per the description of melee attacks (Basic Rules, p. 76; PHB, p. 195):

Used in hand-to-hand combat, a melee attack allows you to attack a foe within your reach. A melee attack typically uses a handheld weapon such as a sword, a warhammer, or an axe. A typical monster makes a melee attack when it strikes with its claws, horns, teeth, tentacles, or other body part. A few spells also involve making a melee attack.

So your Shocking Grasp counts as a melee attack with your touch.