No, because a PC who is dropped to 0 hitpoints is not killed, but unconscious and dying. The Players Handbook takes some pains to distinguish this. Also, Finger of Death has an Instantaneous Duration, while it can take up-to 5 Rounds for a character to finally die.
I am going to paraphrase an answer I made to another question on this same subject.
Death Saving Throws
If damage reduces you to 0 hit points and fails to kill you, you fall Unconscious (see Conditions ). This unconsciousness ends if you regain any hit points.
That itself is pretty-clear: If an attack drops you to 0 hitpoints but doesn't exceed your hitpoint maximum, by RAW it fails to kill you.
Finger of Death
You send negative energy coursing through a creature that you can see within range, causing it searing pain. The target must make a Constitution saving throw. It takes 7d8+30 necrotic damage on a failed save, or half as much damage on a successful one.
A humanoid killed by this spell rises at the start of your next turn as a Zombie that is permanently under your Command, following your verbal orders to the best of its ability.
So, let's look at Mac, a Level 5 Fighter with 44 Hitpoints. Mac, at full health, decides to go bully a high-level Warlock NPC.
The Warlock, not amused at all, casts Finger of Death on Mac, who fails his saving throw and takes 67 (rolling 7d8+30) points of damage. This is enough to send him from 44 to 0, but the damage remaining is 23, which doesn't exceed 44. Therefore, Mac does not die; by RAW, the spell fails to kill him outright, but he is now unconscious.
Now, Finger of Death, an Instantaneous-Duration Spell, works by arcing negative energy through the target. After the spell is over, the negative energy (the same kind of energy which, by D&D official Lore, animates Undead) is no longer coursing through Mac's body. In fact, it has pretty-much dissipated after draining most of his energy. If Mac were to fail his next 3 saving throws, it wouldn't be the spell that killed him: the magic is over and done with. Instead, it is the fact that he can no longer procure enough life energy to continue to beat his heart or breathe which kills him.
That is why Mac would not come back as a Zombie, because there is no negative energy left from the spell to animate him, that expired several turns ago when the spell's duration (Instantaneous) wore-off.