Yes and No
No: So long as you only learn the tools required to get the job done, having a CS degree is not required.
Yes: You may not even learn or study things that are directly related to web development. However what you will learn is how to learn, plus have a much bigger and deeper understanding of entire domain of CS.
Here is the difference:- taken from personal experience, names hidden to protect the innocent ;)
- Developer A (Veteran) - The Experienced Field veteran 20+ development under the hood
- Developer B (Graduate)- Fresh out of University, but with some freelancing projects here and there.
The graduate joins the veteran, initially he is inspired and highly respects the Veteran for his years of experience.
Initially the graduate does how the Veteran has designed things. The young grass hopper has much to learn, and makes the usual 'newbie' mistakes.
As time goes by however the graduate starts to have doubts, he starts to see things that are not right. But he can not undermine the Veteran's influence and authority and experience.
As more time goes by, the graduate starts to not only handle anything that is thrown he's way, buts starts to do it, faster, and with more stability. Everyone soon finds he's code is better maintainable and usually done in half the code of the veteran.
A little further down the line, the Veteran leaves. The graduate finds himself finally in a place to implement things he could not previously.
Projects that would have previously been only 'acceptable' now receive praise. The only concern raised by clients is "We was told this was not possible, how did you manage do it?"
Moral of the story:
You don't need a CS degree to be web developer, but having a CS degree might[1] just make you a better developer overall..
[1] some people are just naturally talented, degree or no degree.