There are two issues involved. The first is the private property status of money substitutes. It’s clear how money certificates are consistent with private property. It’s not clear how fiduciary media are so. People hold (i.e., own) fiduciary media as part of a ready stock of medium of exchange and yet, simultaneously the un-backed portion is lent to other people to use as a medium of exchange. Rothbard thought fractional reserve banking, as it has been practiced, was fraudulent. Take a look at the following article to see the back and forth of the debate with free bankers.
The second issue is what would happen on the market if all the legal privileges for fractional reserve banking were eliminated. Then demand deposits would be highly liquid assets and if banks wanted to stimulate customers to hold such balances by ensuring that the deposits serve as a medium of exchange, they would need to hold very high, if not 100%, reserves. Otherwise non-customers of the bank of issue would not accept the deposits as a medium of exchange. As you note, this view is taken by Rothbard. It was also the view of Mises and Salerno. Take a look at the following piece by Salerno.