How Long Would It Take For The Orbit To Decay Away?
It might take far longer than the age of the Universe itself, but it wouldn’t be arbitrarily stable.
Before we ever measured any Gravitational Waves, in fact, this was the primary method of measuring the Speed of Gravity.
Not for the Earth, but for an extreme system where the orbital changes are easily observed: for a tight orbiting system containing at least one neutron star.
The largest effects will show up for a massive object moving with a rapidly changing velocity through a strong, changing Gravitational Field.
That’s what a Binary Neutron Star system gives us!
As one or both these Neutron Stars orbit, they pulse, and the pulses are visible to us here on Earth each time the pole of Neutron Star passes through our line of sight.
The predictions from Einstein’s theory of gravity are incredibly sensitive to the Speed of Light; so much so that even from the first binary pulsar system discovered in 1980s, PSR1913 +16 (or the Hulse-Taylor Binary), we have constrained the Speed of Gravity to be equal to the Speed of Light with a measurement error of only 0.2%.