Yes, the index of refraction of air does depend on the density of the air, usually expressed in terms of the air pressure rather than the density.
This effect limits the accuracy of displacement measurements by interferometry, particularly when measuring the displacement of a moving object which is producing turbulence (air pressure variations) in the air around it.
The fractional content of water vapor and CO2 in the air also affect the index of refraction measurably.
From some brief web research, there are widely accepted fitting formulas for these effects from Edlen (1966) updated in 1994 by Birch and Downs; and by Ciddor (1996). A presentation from the Canadian National Research Council gives formulas based on Edlen, Birch, and Downs:
Sadly, the individual terms (particularly $x$, $\sigma$, and $f$) are not fully explained, so you'll have to work out exactly what they mean or go back to the primary sources for an explanation.
The US's NIST provides an online calculator based on Ciddor, and some helpful instructions. I also found a page where you can download Python code for calculating the refractive index based on Ciddor.
I don't find any simple formula that gives just the sensitivity of index to pressure, but from the NIST page it seems that a difference in air pressure of approximately 0.4 kPa (standard air pressure being 101.325 kPa) produces an index of refraction change of about 1 ppm (this number likely slightly variable depending on wavelength, temperature, air composition, etc).