It can't be a stronger property, the essence of Turing-completeness is that a Turing-complete automaton can simulate any other Turing-complete automaton, including itself.
Whether an automaton, capable of scale-invariant self-simulation, but not exhibiting Turing-completeness, is even possible, is a very interesting question. I don't know the answer.
Is the ability to do this tied to the fact that Game of Life is Turing-complete, or is there a weaker/stronger property that allows for this?
Also, does this count as a “quine”?