Parameters of Butterworth filter that approximates the amplitude response of a Gaussian filter?
The solution is to implement a FIR filter with the coefficients using the Gaussian response desired. The center dominant tap will set the delay of the filter, which will of course not be zero phase, and the result will be the causal one-pass filter desired. 

Also related, see my post at the link below on how you can implement this filter as a cascade of 2 tap unity gain filters (it doesn't get much simpler than that!).

Gaussian FIR filter with no multipliers?