Is there an equivalent time-domain operation to the spectrum averaging in Welch's method?
Yes, you can average in time lag (autocorrelation) as well as in frequency. You can also combine the two domains, as long as it is reasonable to treat the data as being stationary.

A. H. Nuttall and G. C. Carter, "Spectral estimation using combined time and lag weighting," in Proceedings of the IEEE, vol. 70, no. 9, pp. 1115-1125, Sept. 1982.