The answer is 'no'. In fact, even for a Ricci-flat Kähler manifold $(M^4,J,g)$, the map $W_-(x):\Lambda^2_-(T_xM)\to\Lambda^2_-(T_xM)$ can be any symmetric traceless linear map. Hence, the only constraint on the eigenvalues is that they sum to zero.
This fact was known to Élie Cartan already in 1926, although he did not use such terminology, as it had not yet been invented. What he considered was Riemannian 4-manifolds with holonomy contained in $\mathrm{SU}(2)\subset\mathrm{SO}(4)$ (what we now call a Calabi-Yau metric in complex dimension $2$). He observed that such metrics were Ricci-flat and that the curvature of such a metric consisted entirely of what we now call $W_-$, the anti-self dual Weyl curvature, and he showed that it could take on any trace-free value as above. (See his little book, Leçons sur la géométrie des espaces de Riemann, particularly the last couple of pages.)
Of course, Cartan only considered the local theory, not global examples. However, it is not hard to show that, up to diffeomorphism, the space of germs of $\mathrm{SU}(2)$-holonomy metrics in dimension $4$ for which $W_-$ has a double eigenvalue everywhere is finite dimensional and that they always have non-trivial Killing vector fields. In particular, the Calabi-Yau metrics on K3 surfaces are not of this kind, hence, their $W_-$ has three distinct eigenvalues at a generic point.