It has been shown elsewhere that in a classical spacetime with multiply connected space slices (wormhole spacetime), closed timelike curves can form generically. The boundary between an initial region of spacetime without closed timelike curves and a later region with them is a Cauchy horizon which can be stable against small classical perturbations. This paper investigates stability against vacuum fluctuations of a quantized field, by calculating the field's renormalized stress-energy tensor near the Cauchy horizon. The calculation is restricted to a massless, conformally coupled scalar field, but it is argued that the results will be the same to within factors of order unity for other noninteracting quantum fields. The calculation is given in order of magnitude for any spacetime with closed timelike curves, and then a detailed calculation is given for a specific example of such a spacetime: one with a traversable wormhole whose mouths create closed timelike curves by their relative motions. The renormalized stress-energy tensor is found to diverge as one approaches the Cauchy horizon. However, the divergence is extremely weak: so weak, that as seen in the rest frame of one of the wormhole mouths the vacuum polarization's gravity distorts the spacetime metric near the mouth by only δgVPμν~(lP/D)(lP/Δt), where Δt is the proper time until one reaches the Cauchy horizon and D is the distance between the two mouths when the Cauchy horizon forms. For a macroscopic wormhole with D~1 m, δgVPμν has only grown to lP/D~10-35 when one is within a Planck length of the horizon. Since the very concept of classical spacetime is normally thought to fail, and be replaced by the quantum foam of quantum gravity on scales Δt<~lP, the authors are led to conjecture that the vacuum-polarization divergence gets cut off by quantum gravity upon reaching the tiny size lP/D, and spacetime remains macroscopically smooth and classical and develops closed timelike curves without difficulty. Hawking, in response to this, has conjectured that the spacetime near the Cauchy horizon remains classical until DΔt (which in a certain sense is frame invariant) gets as small as ~l2P, and correspondingly until δgVPμν~1, and that, as a result, the vacuum-polarization divergence will prevent the formation of closed timelike curves. These two conjectures are discussed and contrasted. The attempt to test them might produce insight into candidate theories of quantum gravity.