I don't believe that is possible. I don't think you'd want to save the actual URL you're using for your ajax request anyway....would you? Wouldn't that either be XML, JSON or just an HTML snippet, and not your entire page? Why don't you add an optional argument to your page....like
http://www.domain.com/search/?crit=s...teriaHere...or something....then have a link or button that says Bookmark these results or send these results to a friend...and send them the link above so you load your entire page, and not just the results from your ajax call. Just change searchCriteriaHere each time they do a different search.
Does that make sense?