As "Rai Südtirol" reported on Wednesday, the move led to a heated debate about the limits of political competence.
Several members of parliament labelled the motion inadmissible. They spoke of "questionable interference" and warned of a "dangerous precedent".
Other politicians, however, defended the right of the state parliament to introduce uncomfortable topics.
In the end, a majority of the state parliament declared the motion admissible, but rejected the actual demand for resignation.
The background to the demand is a personnel decision by the Bishop of Bolzano, which caused outrage in South Tyrol.
In September, Muser had transferred a priest to a new pastoral position, despite the fact that he was incriminated in an abuse report.
Following massive criticism, the bishop retracted the decision and admitted that he had made a mistake.