Yes. The analogue of Uhlenbeck’s result holds if you restrict to Kähler metrics, even within a fixed Kähler class: for a residual (comeagre) set of Kähler metrics in a given class, the scalar Laplacian has simple spectrum.

Why the same argument goes through

• Parametrization of the space: Fix a background Kähler form ω0 in the given Kähler class. Every other Kähler form in the class can be written as ωφ = ω0 + i∂∂̄φ, where φ is a smooth real-valued potential and ωφ > 0. For k large, the set of Ck+2 potentials with ωφ > 0 is an open subset of a Banach space (mod constants), hence a Baire space, so “generic” makes sense.
• Uhlenbeck’s scheme: Consider the zero set of the map F(φ,u,λ) = Δgφ u − λu with normalization on u (e.g. ∥u∥ = 1 and ∫u = 0). The key step is to show transversality: at a triple (φ,u,λ) with an eigenvalue of multiplicity m ≥ 2, the derivative of F in the φ–direction must be rich enough so that, restricted to the m-dimensional eigenspace, the resulting m×m perturbation matrix can be made arbitrarily “generic” by suitable choices of φ. Then Sard–Smale yields that multiple eigenvalues are avoided by a residual set of φ.
• Allowed variations are ample: A variation φ ↦ φ + tψ changes the metric within the fixed Kähler class by δω = i∂∂̄ψ; at the level of the Riemannian metric, this produces a J-invariant symmetric 2-tensor h that depends linearly on the complex Hessian of ψ. Locally, by prescribing the 2-jet of ψ, one can realize any Hermitian quadratic form at a point; by using ψ with small support, these variations can be localized. Thus, although we are restricting to Kähler variations (and even to a fixed cohomology class), the space of admissible first-order perturbations h is still infinite-dimensional and pointwise contains all J-invariant symmetric bilinear forms.
• Breaking multiplicities: The first-order variation of the eigenvalue problem along such a perturbation yields a symmetric m×m matrix whose entries are integrals of bilinear expressions involving h and the gradients of a basis of the eigenspace. By choosing ψ supported in tiny, disjoint balls and using unique continuation to pick points where these gradients are in favorable (generic) positions, one can span all symmetric m×m matrices with these variations, exactly as in Uhlenbeck’s argument. Hence one can split any multiplicity at first order, and transversality follows.

Two sanity checks

• Complex dimension 1: A Kähler metric in a fixed Kähler class on a Riemann surface is just a metric in a fixed conformal class. It is known (and can be shown directly by the above scheme) that generic conformal factors give simple spectrum.
• Symmetry does not obstruct genericity: Special Kähler metrics (e.g., highly symmetric ones like Fubini–Study) have large multiplicities, but these are destroyed by arbitrarily small Kähler potential perturbations; the set where multiplicities occur is meagre.

References and comments

• The proof is a straightforward adaptation of K. Uhlenbeck, Generic properties of eigenfunctions, Amer. J. Math. 98 (1976), 1059–1078, replacing the full space of metrics by the Banach manifold of Kähler potentials in a fixed class and observing that the linearized variations cover all J-invariant quadratic forms pointwise. The required unique continuation for eigenfunctions and the perturbation formulae are the same as in Uhlenbeck.
• Closely related ideas appear in work on generic simplicity in constrained families (e.g., within a conformal class), and the strategy is identical: the key is having enough localized parameters to make the perturbation matrix on any finite eigenspace arbitrary. The Kähler-potential parametrization provides precisely that.

So the answer is affirmative: for any compact Kähler manifold and any fixed Kähler class, the set of Kähler metrics in that class whose scalar Laplacian has simple spectrum is residual in the Ck topology (k large), hence dense and generic.