I can't speak for Rothbard since I'm not familiar with all of his writing. Inflation is a monetary phenomenon caused by an over expansion of the money supply. Gold redemption serves as a brake on over expansion of the money supply, it doesn't prevent an increase of credit money. Banks create credit money when they issue loans. The creation of credit was predicated on the presentation of a 'real bill', and real bills doctrine is too much of a tangent to pursue in a post.
Don't confuse yourself with "the fallacy of the beard": one hair doesn't make a beard. Nor do ten. But at some point enough hairs do make a beard. The same with credit expansion. Gold convertibility provided a means to shave the beard.
Besides, how could a bank loan more money than the bank actually itself owned? Would not that bank be insolvent?
Banks have always loaned more money than they have on deposit. That's how fractional reserve banking works, and to cite Mises fractional reserve banking is banking. And it worked the same when the gold standard was in effect. Having no gold standard won't protect banks from insolvency, the problem of illiquidity on technically sound assets still exists.
You can find most of Rothbard’s writings at the Von Mises Institute. Trust me, Rothbard believed in a fixed money supply.
The real issues here, since this is a Ron Paul thread, is what is Ron Paul’s stance on fractionalized banking and the money supply? I believe he opposes, but if you have other information, please provide.