Nope, there are three vaccines, and (in the US) only one kind of document that proves vaccination with them. On the other hand, you'd either have to prove positive infection and recovery (with multiple test results) to someone (whom?) or to standardize some kind of antibody testing which also has to take place soon after the infection, because those antibodies don't last very long. The immunity might last longer, but the kind of antibodies that are modestly easy to detect, don't. But because of the nature of those kinds of tests, every single lab would need to be certified individually and there would need to be another step to actually issue some kind of certificate.

So yes, to me it sounds massively more complicated and expensive than just getting two harmless vaccinations.