Why does this ‘r’ appear only in ‘wash’ and ‘Washington’ without analogous examples? That is, why does this ‘r’ not also appear in similar constructions (like ‘posh’ (which is never pronounced ‘parsh’) or ‘to josh’ (which never allows for ‘I was jorshing/jarshing’))?
In fact, it seems to be the case that you do get /r/ in those contexts and specific words (also, 'squash', etc.). It may be difficult to find them in text (e.g., for 'posh': 'porsh' and 'parsh' are both problematic for the speller), but they likely exist. I was able to find Google hits for 'jarsh'/'jarshing'/'jorshing', for example.
It is also possible that this kind of regularization rule is applied more for high-frequency words than for low-frequency. If 'wash', 'water' are much higher frequency than 'posh', 'josh', then it's reasonable to see fewer instances of 'jarsh'.