Can Gaussian dist. use median/MAD instead of mean/SD?
I am not 100% sure if I got your question, but I will try my best.

By looking at the gaussian probability density function and inspecting the parameters you will see, that there is actually no mean at all. There is only the expected value $\mu$, which is not necessarily the mean of your data, and $\sigma$, which depends on $\mu$.

When you have data and want to fit it to a gaussian, you will see by applying maximum likelihood, that using the mean is quite the best thing you can do to fit a gaussian.

Ideally your mean and median should be close. If not, there are two options. First your data is simply not gaussian. Second you have outliers or errors in your data. Then you can think about cleaning your data first or as a lazy and fast solution take the median.
Otherwise I see no justification to take the median.