No.
If the browser is making the request, the user can spoof the request. Period.
My web application knows how to correctly use the API
That's good, but that's leading you down the path of assuming client-side functionality executed as intended. Never make that assumption.
I might not have thought of every single validation known to man to prevent bad data from being sent through the API
This statement leads me to believe that the API itself is more complex than it needs to be. The best thing you can do is simplify. It's difficult to be more specific without seeing specific code, but API requests should be fairly simple and straightforward and the same techniques to prevent malicious code from getting through should be applied universally. The same general rules apply here as in any web application interaction...
- Never trust anything that came from the client
- Never assume client-side code executed as intended
- Never execute input as code, always treat it as a raw value
- and so on...
As you mention toward the end, you've already taken care of authentication and authorization for the requests. Given that, if User X is permitted to make a given API call, then what you're essentially asking is, "How do I allow User X to make an API call without allowing User X to make an API call?" The server can't tell the difference. A request is a request.
Sure, there are things you can try, such as always including some custom header in requests made from code. But anybody can inspect that request and spoof that header. The user's browser isn't part of your application and isn't under your control.
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