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So could I cast Vampiric Touch (this last 1 minute) and using my unarmed attack bonus do Vampiric Touch damage + unarmed strike damage? I know I would only gain health from the Vampiric portion of the damage.
If I can't would it be a reasonable house rule?
To add some specifics we're talking about STR 14 INT 18
which means if it hits it will do 3D6+3(instead of a flat 3D6) and my attack roll will be +5 instead of a +7.
So could I cast Vampiric Touch (this last 1 minute) and using my unarmed attack bonus do Vampiric Touch damage + unarmed strike damage?
Nope. Vampiric Touch is a Cast a Spell Action, making an Unarmed Strike is an Attack Action. Attacks with Vampiric Touch use your Spellcasting Attack Bonus, as it is a spell. The subsequent attacks with Vampiric touch are likewise specific actions that precludes you from making an Attack Action with your unarmed strike.
If I can't would it be a reasonable house rule?
Also no, an extra 3d6 necrotic damage along with your unarmed strikes is a huge imbalance, that's like an instant Sneak Attack as a 5th-level rogue with no need for ally adjacency or Advantage, not to mention you can up-cast the spell to keep up with the rogue. It's additional damage (10.5 average) is also better than a Paladin's Divine Smite (9 average) per hit, and it consumes only 1 spell slot that you can then use up to 10 times. Oh, and it heals yourself on top of all of that.
Sacrificing your +2 attack bonus for +3 increase in damage might sound reasonable to you now, but there are too many ways to improve those bonuses including picking up certain feats, class features, and magic items. Soon, your houserule will become overpowered.
When asking if it would be a "fair" houserule, you have to consult with your DM and the rest of the party considering that the stackexchange does not know your current situation and the relevance behind vampiric touch for your character or playstyle (due to the fact that the rules can be bent to suit the needs of the players if the DM rules that it would be much more enjoyable that way).
Regarding the direct melee spell attack rule however, when referring to the PHB;
Your attack bonus with a spell attack equals your spellcasting ability modifier + your proficiency bonus.
Due to the fact that it does not dictate otherwise the general assumption is that you can't simply use your unarmed attack modifier rather than your spell attack modifier due to the fact that what you are using is a spell and not, say, your fist.
This is further backed by the spell description of Vampiric Touch directly stating that;
The touch of your shadow-wreathed hand can siphon life 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.
Hopefully that was helpful enough to suit your needs.
Using Vampiric Touch takes an action, so does an Unarmed Strike, you can use one or the other.
Vampiric Touch is a horribly underpowered spell. For a 3rd level slot it does less damage than a cantrip: Firebolt deals on average 11 points of damage, while Vampiric Touch does 10.5. The healig it provides is mostly offset by the excess damage you will receive by being in melee range. Using up your Concentration makes it even worse.
For you currently it is mostly not much stronger to attack with +5 and +3 to damage than +7 as attack bonus and +0 to damage. It provides the same DPR against AC19, and weaker for higher ACs.
Even if your STR is maxed out, as long as you do not attack more than once per round with Vampiric Touch, this change is barely enough to make it competitive.
Considering it is trivial to get Advantage and even easier to have one of your allies into position, the comparison with a Rogue shows that spending a 3rd level slot and your Concentration is quite a steep price for something that Rogues can do every round and even out of their turn.
