If? It's already happened: http://en.wikipedia.org/wiki/Mycin#Results (and this was in the 70s).

Then why isn't there a website to get medical diagnosis on a self-serve basis? (I'm not asking rhetorically, genuinely curious)

The FDA, mostly. They have a multi billion dollar industry to keep in business.

Yeah, in a very limited set of circumstances, and still with a pretty significant miss rate.

But the day will come when the machines are near-perfect.

And people already trust computers more than they did in the 70s. Perhaps not if you ask them directly, but the reality is that they do, with many areas of their lives, and without a second thought.