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Recent spectral graph sparsification techniques have shown promising performance in accelerating many numerical and graph algorithms, such as iterative methods for solving large sparse matrices, spectral partitioning of undirected graphs, vectorless verification of power/thermal grids, representation learning of large graphs, etc. However, prior spectral graph sparsification methods rely on fast Laplacian matrix solvers that are usually challenging to implement in practice. This work, for the first time, introduces a solver-free approach (SF-GRASS) for spectral graph sparsification by leveraging emerging spectral graph coarsening and graph signal processing (GSP) techniques. We introduce a local spectral embedding scheme for efficiently identifying spectrally-critical edges that are key to preserving graph spectral properties, such as the first few Laplacian eigenvalues and eigenvectors. Since the key kernel functions in SF-GRASS can be efficiently implemented using sparse-matrix-vector-multiplications (SpMVs), the proposed spectral approach is simple to implement and inherently parallel friendly. Our extensive experimental results show that the proposed method can produce a hierarchy of high-quality spectral sparsifiers in nearly-linear time for a variety of real-world, large-scale graphs and circuit networks when compared with the prior state-of-the-art spectral method.