No, characters do not automatically fail a death saving throw upon dropping to zero hit points.
According to the basic rules, under "Dropping to 0 Hit Points" in the Combat chapter, a character either dies instantly or falls unconscious when they drop to 0 HP. If they're unconscious, then during the following turns they must succeed three death saving throws to stabilize, or fail three death saving throws to die.
On your third success, you become stable (see below). On your third failure, you die.
Normally, a creature can fail death saving throws for one of two reasons:
They begin their turn with 0 HP. They attempt the death save, and fail. This counts as 2 failures if the result was a natural 1.
They are already at 0 HP, and then take damage from another attack or effect. This counts as 2 failures if the damage is caused by an attacker's critical hit.
The rules do not say the character automatically fails one save when they first fall unconscious. Otherwise the character would die after only two failures, and there would be no reason to count the third failure. Therefore, your DM's ruling does not match the basic rules.
It's possible that your DM is misinterpreting the following text:
Damage at 0 Hit Points. If you take any damage while you have 0 hit points, you suffer a death saving throw failure.
... But this rule only occurs while the creature has 0 HP, so the initial damage that had dropped them to 0 HP would not qualify.
Note that a creature cannot be below 0 HP. Zero is the minimum. If a creature has (or falls to) 0 HP, and the remaining damage equals or exceeds their HP maximum, then they die. But they cannot have negative HP.
Also note that exceptions may exist. For example, there could be an especially nasty creature whose attacks inflict one failed death save upon dropping a creature to 0 HP. Such a feature would need to explicitly mention this behavior, otherwise it would function as per the basic rules.