Does casting Inflict Wounds while concentrating on Vampiric Touch also siphon damage?

Question

Vampiric Touch says,

The touch of your shadow-wreathed hand can siphon force from others to heal your wounds. Make a melee spell attack against a creature within your reach. On a hit, the target takes 3d6 necrotic damage, and you regain hit points equal to half the amount of necrotic damage dealt. Until the spell ends, you can make the attack again on each of your turns as an action.

Scenario:

• Round 1 you cast vampiric touch and make a melee spell attack for 3d6 necrotic damage.

• Round 2 while still concentrating on vampiric touch you cast Inflict Wounds as a 3rd level spell for 5d10 necrotic damage.

Being that Vampiric Touch is still functioning, you are making a melee spell attack, and it is dealing necrotic damage... Does the necrotic damage from Inflict Wounds siphon health?

(Obviously the 3d6 and 5d10 wouldn't stack because of action economy, etc.)

I am sure that a player could find a better use of two 3rd level spell slots, but I have taken up a kind of awkward place as an arcane healer/tank for the group and every bit of preservation helps.

Answer 1

No

The spell says you make an attack, and describes what happens when the attack hits. It is not a general buff, it does not apply to any other attacks, it does not apply to any other necrotic damage.

If you read below (emphasis mine), you will see it says you make "a" spell attack, on hit the target takes damage and you gain hp equal to half "the" amount of damage dealt.

Make a melee spell attack against a creature within your reach. On a hit, the target takes 3d6 necrotic damage, and you regain hit points equal to half the amount of necrotic damage dealt. Until the spell ends, you can make the attack again on each of your turns as an action.

You make an attack, and then gain the ability to make that attack again. The effects don't apply to any other circumstances or attacks.

Answer 2

The touch of your shadow-wreathed hand can siphon force from others to heal your wounds. Make a melee spell attack against a creature within your reach. On a hit, the target takes 3d6 necrotic damage, and you regain hit points equal to half the amount of necrotic damage dealt. Until the spell ends, you can make the attack again on each of your turns as an action.

Emphasis by me, meaning that vampiric touch refers to a specific attack, the 3d6 attack.
You still get to concentrate on vampiric touch so you can use it on another turn, but you do not get to drain using another spell.

Answer 3

The source of your confusion comes from two factors that are in the core of 5e:

1. Effects do not stack unless they say so.

2. The characters are constrained in combat by the number of actions they can take. This is called in the meta as "Action Economy."

So, You can't combine the attacks from two spells' unless they say they can, the first principle. Doing so would allow you to "combo" two actions in one, that goes against the second principle.

The effect's wording is very important. Check the special action that vampiric touch allows:

Make a melee spell attack against a creature within your reach. On a hit, the target takes 3d6 necrotic damage, and you regain hit points equal to half the amount of necrotic damage dealt. Until the spell ends, you can make the attack again on each of your turns as an action.

This tells us a lot of things.

1. You make A melee spell attack. One. To make attacks, you take the Attack action. And yes, Attack action is different from an attack

2. That attack is a melee spell attack.

There are other coinstraints in the text that are irrelevant now. I want you to check the first word of the sentences quoted above.

Make an attack.

See that it does not says When you use a melee spell attack. This wording would allow you to add the effect on another attack. For an example of damage that goes in addition to another attack, check the rogue's sneak attack:

Once per turn, you can deal an extra 1d6 damage to one creature you hit with an Attack if you have advantage on the Attack roll. The Attack must use a Finesse or a ranged weapon.

So no, you can't combine these two spells.