Can this be considered a complete sentence?

No. It's ungrammatical.
The rule of There-Insertion requires some adverbial -- of place, time, or circumstance.

Except, let it be said, as usual, with be, which it most commonly occurs with.
There are some common constructional usages with be that don't need adverbials,
e.g:

• Enumerating lists: There's holmium, and helium, and hafnium, and erbium.
• Asserting existence: There is a number which is the square root of -1.
  (often with stressed be).

But other There-Insertion verbs like live require an adverbial. That's the function of story-initial phrases like Long, long, ago or Once upon a time or In a castle above the city -- they establish the time, place, or circumstance of the existence.

One could of course say

• A princess named Gretchen lived.

But that's similarly incomplete; put an adverbial in

• A princess named Gretchen lived long, long, ago.
• Long, long, ago, a princess named Gretchen lived.

and it's still weird, because English does not prefer indefinite subjects with existential verbs.
That's what There-Insertion is for; it inserts the dummy subject there.

• Long, long, ago there lived a princess named Gretchen

Sure. That's how fairy tales often begin.