No, it's not true
As reported here, the new hospital does have a prayer room with washing facilities, but the prayer room isn't "exclusive" to Muslims. It merely accommodates Muslims, who wash as part of their five-times-daily prayers. Those involved in the design say that if Christians or others had a similar requirement, they would be accommodated in the same way.
Members of Chaplaincy Services SA and the Uniting Church SA worked on the "inclusive sacred space" in the new hospital. In the article, Uniting Church SA Moderator Rev Sue Ellis said:
"I’m disappointed that the work of faithful chaplains has become politicised when people who are in the RAH need loving support.
"Our chaplains are guests in the space held by patients at any hospital. They believe the best way to minister the Gospel of hope and healing in the social hospital context is by focussing on the patient and their needs. They follow the example of Jesus, who noticed the need of the vulnerable and asked about them, before he offered healing.
"The Christian church has a long history of supporting and establishing hospitals. They provide a space for us to show God’s mercy, grace and kindness to all people in need, including people from other – or no – spiritual traditions. In focussing on what helps give life and hope to the one in need, the church has always respected the rituals of other faiths.
"The prayer space with washing facilities provided in the new RAH serves an essential purpose for people of Muslim faith. If practicing our faith required such elements they would be also be provided. This is because the focus is on the need of the patient, not on the institutional religion."
(Henry didn't seem to want to convert his comment to an answer, so I'm posting this myself based on his research.)