According to this article, nacre is composed of aragonite ($\mathrm{CaCO}_3$) and organic matrix (chitin, proteins ...) with a mass ratio $95 \% : 5 \%$. In the supplementary file, they provide densities for each component, namely the density of aragonite $\rho_a=2.95 \; \mathrm{g/cm}^3$ and the density of a generic protein $\rho_p=1.35 \; \mathrm{g/cm}^3$. From this data, we can _estimate_ the density of nacre:
$$\rho_n \approx \frac{1}{\frac{0.95}{\rho_a} + \frac{0.05}{\rho_p}} = 2.78\; \mathrm{g/cm}^3.$$