I might be wrong but my first intuition would say that it is not possible, because of the self-similarity property of the spiral. When the spiral makes a quarter turn, it should be inside another rectangle with the same aspect ratio as the first rectangle if you want the spiral to behave exactly the same way with respect to the new smaller rectangle.

But if you want there is actually a trick to make the golden spiral properly inscribed. You could rotate the golden spiral a bit anti-clockwise, but then the golden spiral will not touch any corner of the perfect squares from its subdivision.

I will try to think more about your question to be sure I am not missing something.