Umm, only if you don't do anything but serve static HTML.
Execute anything "custom" on the server (like JSP, generated pages, old-style CGI, etc.), and you have to design it specifically for the dirty environment or THAT can be used to exploit. It's an enormous hole in the firewall, that can be arbitrarily large if the designer screwed up and let a buffer overflow (for instance). It's surprisingly easy to get access to a command interpreter (including the Java virtual machine) through that type of hole. The issue is not necessarily the visible program -- these days, most of that has been offloaded to the client -- but rather the server code assembling it.
This is not something to be done casually. You MUST be aware of the implications of what you are doing here, and it is far worse than a simple desktop.
As for Microsoft quality, it's VERY clear where the development effort has gone -- it's in "bling," rather than the more important nuts and bolts such as performance and security. It's more important to have windows fade in a pretty fashion than to have a registry that doesn't corrupt. And external review is simply not possible because Microsoft keeps their several-million LOC source code well hidden. And the size is unmaintainable with an infinite number of software engineers.
The example of Google has been given. Those guys run an ARMY of programmers on their webservers, all built upon Apache running on Linux. Not Microsoft. For a reason.