Do you think web designers need an understanding of the fundamentals of art?

As a web programmer with a degree in Fine Arts... YES. They do.

When you really get down to it, the web is about using visual media to communicate ideas. So is Art. Likewise, both disciplines have a heavy technical component. Both have mixed audiences... most people care about the tech side, some just want to experience the final product. Both disciplines also start with a blank page, and build it into the final product. The final products of both disciplines are greatly improved when the audience is engaged, and can be revolutionary when the audience collaborates.

So do web designers need to understand art? Yes!

I agree - I think it's super important. It also helps you be more creative as you have more tools for 'seeing'.

Fundamentals? Mostly yes. As much as web design "degrees" force their students to know? Mostly no.

I say the latter because a huge majority of teaching materials that attempt to cover the web are basically just fluffed up materials for print designers with HTML and CSS tossed into the mix.

I don't think they NEED to have an understanding but I think it makes a major difference. The best designers I've worked with have an understanding of style and aesthetics beyond their craft, extending to art, fashion, cars, cinema, etc.