Is a melee spell attack also a melee attack?

Absorb Elements says:

The spell captures some of the incoming energy, lessening its effct 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.

Shocking Grasp 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 on a D6 Fire to my next shocking grasp attack?