It does not rely on the desktop environment. What it does rely on (at least in case of graphical-mode browsers like Chrome, Opera or Firefox) is the Xorg server, together with its libraries1.
Once you have the core dependencies of a browser installed (including the X server), you can run it under a minimal environment either by:
- using a very minimalistic window manager (WM), such as
- directly running the browser from your
For more examples, have a look at the question about building a minimal, browser-only distribution and the example looped browser invocation I've suggested (that's the "no WM" option).
There's one caveat, though. My limited experience with Ubuntu shows that it is very... Hmm, let's say "generous" - when it comes to calculating dependencies, especially because the default APT configuration auto-installs recommends in addition to absolute dependencies. That is to say, the standard package management tools assume a demand for rich functionality and desktop integration - which leads to many dependencies. So, the bottom line can be that they would probably force you to installing large parts of some DE as a dependency for, say, Chrome. To go about this, you might need to dig deep into some package-management related configuration or even be forced to install the browser independently (not through the package management tools) - which can in turn create mess... You'd be definitely better off using other distribution for this, such as Arch or Gentoo, that are more conservative and customizable in terms of your expected functionality.
1In a very limited way, this can work even without X, using a replacement graphical libraries, such as DitectFB.