I somewhat disagree with Basiclife's answer; if you have any code inside something like
if (IsPostBack) {
it's not going to be equivalent, ie the code is going to be executed if you're just setting the URL. Also, controls keep their state across postbacks but are freshly initialized if you're calling the URL again. This is due to ASP.NET trying to emulate a "normal" application, so the way to make sure a normal call and a postback have the same effect might result in "de-ASP.NET-ing" the entire page.
I'm not sure if what you want works. There probably is a way. But I heavily suspect there's a better way of doing this. If you get a postback for free, and can transmit data, why is it crucial that the data shows up in the URL, instead of being comfortably posted? I can see how you want a page to respond to a URL parameter, and how you might want to change the same parameter later on based on what's happening on that page, but since you always know you're posting back, you can eg override that URL parameter in that case, by something you're posting back. This doesn't sound so nice, but it might actually be less messy. Particularly since you seem to have a reason to not abandon the postback at all (otherwise you could just use a link, right?).