The mean velocity profile (MVP) in thermally stratified atmospheric surface layers (ASLs) deviates from the classic logarithmic form. A theoretical framework was recently proposed (Katul et al. Phys. Rev. Lett., vol. 107, 2011, 268502) to link the MVP to the spectrum of turbulence and was found to successfully predict the MVP for unstable stratification. However, the theory failed to reproduce the MVP in stable conditions (Salesky et al. Phys. Fluids, vol. 25, 2013, 105101), especially when ζ >0.2 (where ζ is the atmospheric stability parameter). In the present study, it is demonstrated that this shortcoming is due to the failure to identify the appropriate length scale that characterizes the size of momentum transporting eddies in the stable ASL. Beyond ζ ≈0.2 (near where the original theory fails), the Ozmidov length scale becomes smaller than the distance from the wall z and hence is a more stringent constraint for characterizing the size of turbulent eddies. An expression is derived to connect the Ozmidov length scale to the normalized MVP (ϕm), allowing ϕm to be solved numerically. It is found that the revised theory produces a prediction of ϕm in good agreement with the widely used empirical Businger-Dyer relation and two experimental datasets in the stable ASL. The results here demonstrate that the behaviour of ϕm in the stable ASL is closely linked to the size of momentum transporting eddies, which can be characterized by the Ozmidov scale under mildly to moderately stable conditions (0.2<ζ <1-2). [ABSTRACT FROM AUTHOR]