The FDA might shut down the independent App developers but I don't think the would shut down the Apple Watch for being able to run those Apps.
These studies cost tens or hundreds of millions of dollars, take months or years to run, and then the process of getting the FDA to approve your product takes more millions of dollars and more years.
I don't think a breast cancer screening study would cost hundreds of millions. You could go to a hospital that does breast cancer screening and let every patient scan themselves with a device while they are in the dressing room.
Then you can gather the statistics of whether the software can reliably produce the same diagnosis as the existing technology.
This is entirely incompatible with quickly evolving technology in a market with many small / new players trying out various ideas.
Yes. Especially at the beginning you would have most of the software being developed by the company who sells the device. Business wise that means it might be very worthwhile for a company who develops this technology because they might have a monopoly on the marketplace.
I don't think a breast cancer screening study would cost hundreds of millions.
That's the wrong analogy; we're talking separately selling a device that tells you some metric, not a diagnosis like breast cancer. So presumably you have to compare it against some existing way to measure that metric.
I don't know how much testing & FDA approval normally costs for nontrivial "metric only" devices (e.g. glucometers), but on the outside view, I doubt it's cheap as you say, because everyone in the system has incentives to raise the price, just like...
Apple's iPhone 7 Plus decided to add another lense to be able to make better pictures. Meanwhile Walabot who started with wanting to build a breast cancer detection technology released a 600$ device that can look 10cm into walls. Thermal imaging also got cheaper.
I think it would be possible to build a 1500$ device that could combine those technologies and also add a laser that can shift color. A device like this could bring medicine forward a lot.
A lot of area's besides medicine could likely also profit from a relatively cheap 3D scanner that can look inside objects.
Developing it would require Musk-level capital investments but I think it would advance medicine a lot if a company would both provide the hardware and develop software to make the best job possible at body scanning.