Didn't have to pay to use CUDA but it was a requirement if they wanted PhysX they'd also have to support CUDA - which would have competed with AMD's own compute platforms (not that anyone ever remembers let alone uses AMD's compute systems) so not surprising they wouldn't accept those terms but does change the facts a little. Even with the switch to Open CL though you can see why they wouldn't want to prop up a competitors technology like that - though I think it would have actually bit nVidia in the rear in the long term as AMD's cards would have been quite attractive for certain areas of compute and playing silly games subsequently to try and sabotage their performance or pull support would have shown nVidia in a very poor light within the industry.
The developer of a CUDA driver by AMD wasn't a license requirement but a functional requirement because PhysX is built on top of CUDA. For AMD to support PhySX they would also have to support CUDA, and nvidia doesn't really mind who supports CUDA because it is license-free and public domain standard, owned by Nvidia.
AMD simply wanted their own alternative solution to work, the same as freesync and tressFX.