My prediction is that News Corp content is not so compelling as to shift behavior of most users. Yes, it will shift some behavior, as there are some people who will want the more inclusive search engine for news (Bing) than the less inclusive (Google). But will it shift enough users globally to make a difference? I suspect no. But I confess I do not know the economics of search engine-dom to know the value of switching some fractional number of users from one source to another. I also don't know how much News Corp derives from ads on its web pages driven there by Google, and whether losing some percentage of those will create problems for them.
Well, it'll be interesting to watch other people gamble.