I do not think it will work. If, just for example, Amazon decide to copyright their API and keep their business in USA for copyright protection, another company, who decides to replicate Amazon's API fully will just do it in Europe. Customers probably will not care, since it is internet. Only possible downside here - latency between USA and Europe.
Although it should be borne in mind that the reciprocal protection of copyright works conferred by the Berne Convention should still technically apply meaning that, to use your example, Amazon could still take action against infringement of their API in whatever country was also a signatory to the Berne Convention.
