Yes, the browser works in very much similar fashion. You can try requesting a url in Chrome, possibly the one you tried in Fiddler. The Network Log of chrome would show you two requests.
The RFC description of HTTP status code can be read over here,
Quoting from there only, regarding the 302 status code:
RFC 1945 and RFC 2068 specify that the client is not allowed to change the method on the redirected request. However, most existing user agent implementations treat 302 as if it were a 303 response, performing a GET on the Location field-value regardless of the original request method. The status codes 303 and 307 have been added for servers that wish to make unambiguously clear which kind of reaction is expected of the client.
When a server responds with a 302 status code, it send back the newer url (to which the current requested old url is to be redirected) to the requesting user-agent (likely a browser). Now, as per the RFC document, the user agent must not request the newer url for 302 status code. Yet most of them do make a second request.
Hope-this-helps.