Is there an equivalent time-domain operation to the spectrum averaging in Welch's method? In Welch's method a time-series is segmented and then the subsequent FFTs of the segments are averaged. Is there an equivalent time-domain operation?

I guess doing the the Welch estimate and producing an impulse response using an FIR filter might do what I want, but I'm looking for a way to do it in the time-domain directly (getting rid of noise, keeping signal properties)...
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.