User Peter has done some testing and research and posted his results on the official forums.
Surprisingly, it seems that Vitality does not impact minion health! Instead, strength and intelligence provide the primary means of survival to your minions:
I am testing my Gargantuan against a Walking Corpse (fat zombie) in nightmare mode, Act 1.
I try to use gear that has only one stat, strength/dexterity/vitality/intellect. So if I test Strength, I will have 500+ Strength, and the rest of the stats will be at 40 (base) unless otherwise specified.
Acronyms
DR = Damage Reduction from Armor
PR = Physical damage reduction form Physical Resistance
Test Results
No gear
Gargantuan dies in 4 hits.
Strength test (Strength increases armor)
545 Strength
1729 Armor (51.17% DR)
11 Physical Resistance (6.03% PR)
Gargantuan dies in 10 hits.
Dexterity test
535 Dexterity
1135 Armor (40.75% DR)
11 Physical Resistance (6.03% PR)
Gargantuan dies in 8 hits.
Vitality test
574 Vitality
39.78% DR
11 Physical Resistance (6.03% PR)
Gargantuan dies in 8 hits.
Intellect test (Intellect increases physical resistance.)
566 Intellect
1164 Armor (41.28% DR)
57 Physical Resistance (25.60% PR)
Gargantuan dies in 10 hits.
Physical Resist test
173 Intellect (from my weapon, I cannot find weapon with PR stat)
1105 Armor (40.11% DR)
141 Physical Resistance (46.12% PR)
Gargantuan dies in 17 hits.
Summary
Gargantuan benefits from armor (strength) and physical resistance (intellect). The damage reduction was very noticeable.
Vitality and Dexterity do not benefit pets. My tests show the gargantuan taking 8 hits to die only because of the extra armor from the gear, not from the stats.
Update:
Lylirra, Community Manager for Blizzard, has indicated that they are looking at pet durability, and they are considering changing minion health to scale off of vitality.