I don't see this happen, at least not in the nearby future and maybe even never. The point is that people want a native experience and likely the web will never be able to give such an experience. I think webapps will always be something comparable to Java apps a few years ago, i.e. work on every operating system and try to integrate the look & feel, but not 100% "get it".
For our start-up we would like to share a big part of the codebase between platforms though, so this year we'll be looking towards replacing our existing codebase with a Mono (C# / .NET) codebase. At least with this solution we should be able to:
a) give users of our apps a native experience.
b) share a big part of our code between Android & iOS apps.
c) be able to expand easier to Windows Phone if the users of this platform increase and we think it's viable or if there's a demand from our clients to move our platform to Windows Phone.
Disclaimer: I'm a Mac user and I've always hated apps that didn't feel 100% native on my preferred platform. Perhaps to me the experience can be explained as a sort of "uncanny valley": http://en.wikipedia.org/wiki/Uncanny_valley