Justification isn't the only way to make text readable.
Line length and leading impact readability a whole lot more than justification. A reasonable paragraph length and indentation can also give you much of the effect of ragged right margins. What ragged right margins gives you is a "silhouette" to navigate by, but paragraphs with indented first lines will give you a similar visual aid and if the line is too long or the lines are to tight, the ragged right margin is not going to help a whole lot.
So full justification can make your text a bit less readable, but proper settings otherwise can offset that. On the other hand, full justification creates a more ordered and enticing text to read. It clearly outlines boundaries and makes the product look neater.
Websites often have ragged right margins, but for more reasons than just readability. First, HTML and CSS lack options for easy type setting. Programs such as InDesign can balance kerning, font width, spaces and word breaks to create a nice looking full justification, but CSS is much more crude and cannot. Also, a website must be assumed to be of adjustable width. When adjusting the width of a text with full justification, the text will jump around and look messy for a while. It may seem like a small thing, but it's a consideration that is made when making websites.
Layout is often a balance between looks and readability. Font choice, margin size, aspect ratio and so on are all part of that. On the whole, justification is a large part of the looks, but only a small part of the readability.