Technically Java applets (or custom browser plugins) should be able to work as well, but I don’t see much of an advantage over Flash (well, probably compatibility with 64-bit browsers). Certainly Flash is more common.
Java's security model is meaning a HUGE Java applet to do this. And a custom plugin requires that you get people to install it. Works if you're Google (that's how they do Voice in Gmail), but probably not if you're Joe's Bait Shop wanting to add a click to call application.