Dr. Semmelweis Ignác Fülöp was Hungarian of German descent. It would have been "mossa kezét" (most likely using the formal third-person). Since he was not churlish, he would have approached such statements as a request, not an order.

He worked in Austria and published his work in German, so it seems highly unlikely he would have said it in Hungarian.

Also, contrary to your claims about churlishness, the article states multiple times that he was a rather rude man