No. LIGO can not measure the expansion of the universe.
LIGO only detects a specific class of distortions of the space-time, which at least are of spin-2 and with a frequency of the order 100Hz. Gravitational waves are spin-2 (quadrupolar) modes, and the one that was observed this time has the dominant frequency from 35Hz to 150Hz, which is in the sensitive range of LIGO. But the expansion of the universe is a spin-0 (isotropic) mode and of the frequency zero, which can not be detected by LIGO. LIGO uses an interferometer to measure the change of the difference between the lengths of its arms, so it will produce no signal if both arms are stretched equally.
The sensitivity of LIGO is frequency dependent. The best sensitivity $10^{-21}$ is achieved only around 100Hz, and for extremely slow processesextremely slow processes like the expansion of the universe (corresponding to 0.1nHz), LIGO basically has no sensitivity. So even if one just wish to use the distance sensitivity, LIGO still can not be used to detect the expansion of the universe, due to the loss of sensitivity outside its frequency range.