We report a study of the basic characteristics of laser polishing of fused silica with a protocol that is particularly suitable for surface smoothing of micro-optic elements fabricated by a laser ablation process. We describe a new, to our knowledge, approach based on scanning a highly controlled small size laser beam and melting areas of tens to hundreds of micrometers of glass using a computer-controlled raster scan process, which does not require beam shaping, substrate preheating, or special atmospheres. Special test samples of silica substrates with prescribed spatial frequency content were polished using a range of irradiation conditions with the beam from a well-controlled {\text{CO}}_2 laser operating at a wavelength of 10.59\mu \text{m} . An analysis is presented of the laser-generated reduction in surface roughness in terms of measurements of the spatial frequency characteristics, and the results are compared with the predictions of a simple model of surface-tension-driven mass flow within the laser-melted layer. This technique is shown to be capable of smoothing silica surfaces with \sim 1\mu \text{m} scale roughness down to levels <1\text{\hspace{0.17em} nm} with no net effect on the as-machined net surface shape, at realistic production rates without a preheating stage, and with noncritical residual stresses.