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Short answer: No.

Long answer: It would be impossible for them to distinguish between content that is safe to reproduce publicly. Judging from how they reacted to the ToS and other dings against their public image, they are working too hard to protect that image than to risk something like this nonsense. And lastly, they have had this power with Google Toolbar forever and haven't done it yet.

Yeah. It's an interesting concept, though: getting around password barriers is by necessity a challenge of having a search engine, and it might limit engines from seeing actually useful content.
It's up to the web app developers to expose the information they want indexed. Google absolutely should not start indexing private data in their public index, and I seriously doubt they're that dumb.

One thing I could see them doing is having a separate opt-in service for web apps, something akin to OS X's Spotlight or Google Desktop Search.

I think that's what the authors meant by it. In the sense that your information should show up in your google search results, similarly to how GDS works. It's of course unfeasible, and probably illegal, to make that information available to the world at large. But if you have access to password protected content, there is no reason for it not to be visible to you when you google for it.
"But what if Google could convince users of the value of indexing web app data without identifying the individual user..."

I'm not sure what they're trying to say, but they don't seem to be talking about indexing your own private data for you.

How can password-protected data not be user-specific? If the web application writers want to make certain data public, they already do this - for example, my Facebook profile is indexed by Google, but the messages I see when I log in aren't.

I don't think google will index password protected web and display it in public domain, however google might keep these visiting data as one datasource of personal information mining, and it will be made use of in google ads production.
It already searches the password protected web (via your history), why would you want to index it?