  Let $L$ be a non-negative self adjoint operator acting on $L^2(X)$ where $X$
is a space of homogeneous type. Assume that $L$ generates a holomorphic
semigroup $e^{-tL}$ whose kernels $p_t(x,y)$ have Gaussian upper bounds but
possess no regularity in variables $x$ and $y$. In this article, we study
weighted $L^p$-norm inequalities for spectral multipliers of $L$. We show sharp
weighted H\"ormander-type spectral multiplier theorems follow from Gaussian
heat kernel bounds and appropriate $L^2$ estimates of the kernels of the
spectral multipliers. These results are applicable to spectral multipliers for
large classes of operators including Laplace operators acting on Lie groups of
polynomial growth or irregular non-doubling domains of Euclidean spaces,
elliptic operators on compact manifolds and Schr\"odinger operators with
non-negative potentials on complete Riemannian manifolds.
