The short answer is no, you don't need it. But it's a bit more complicated than that.
Java isn't used that much on webservers. It's popular, I believe, for certain large scale enterprise and financial systems (
Java EE stuff). Common languages for websites are PHP, Flash / ActionScript, JavaScript and perhaps C# / VB.NET and Perl.
If you're happy coding in C#, PHP, C++ or similar and just don't like Java then you're probably fine blowing it off. But I suspect from your question that you're not.
Unlike (say) PHP, Java is actually quite a nice, clean language, and it's a good one in which to learn to program. It will teach you about
object oriented programming, it will force you use
static typing (which is largely a good thing) and it will introduce you to some
useful syntax that's common to quite a few popular languages today (C#, JavaScript, C++).
A lot of web designers don't know how to program. They may know enough to adapt someone else's script for their site or write a couple of lines of jQuery, but they don't get much beyond that. And I guess it's enough, but it does limit their options. After you've learned to program - in almost any language - you'll find it much easier to pick up new languages because you'll already understand roughly how they work.
It really depends on what you want to do with your web work. If you want to code the back end of websites, as opposed to just designing the front end, I'd consider sticking with it. Not because you'll end up using Java, but because you'll have a much better idea of what you want to do when you start writing PHP, Ruby or C#. And don't forget that there's a lot of increasingly complex JavaScript out there now too thanks to all of these web apps. On the other hand, there may be another course that you feel is more useful for your future career.
I know that's all a bit hand wavy, but it's the best I can do without knowing you, your background or your aspirations.