Netscape tried to do it, with lots of manpower thrown at it, and when browsers had way fewer features than today, and failed. So I'd say look for some other project that doesn't compete with powerful, free, native-code products on the market.
Ofcourse it's possible in principle - you can make anything with Java.
Swing already contains classes that can render HTML (see the package javax.swing.text.html) so you won't have to write your own HTML renderer and parser from the ground up, although the Swing classes are most likely not as sophisticated as the major web browsers.
Java also has classes for reading data over HTTP (see java.net.HttpURLConnection).
So it's probably not too difficult to make a simple browser with Java.
As a matter of historical perspective, I would like to point out that when Java was first introduced, the most complex and powerful Java application was in fact the "Hotjava" browser. Google search for "hotjava browser" will get you to the source code.
It was pretty impressive at the time, especially the molecular model applet that let you rotate a "ball and stick" model in real time with the mouse.