Update: if I run The Gimp as sudo from the CLI, I get the same warning, but not
the segmentation fault, and The Gimp does launch. Trying to run The Gimp from
the applications launcher or "Open With:" results in a little disk activity and
then nothing.

You don't say which version of the GIMP. Sometimes people have outside
sources such as Getdeb or a PPA enabled and they are using a version that is
not stable, 2.7 for example. I use the GIMP almost every day on 11.04 with
no problem. I am using version 2.6.11 which is the regular one in the
repositories. Segfault could be caused by a compiling error in which case
getting a fresh version may fix it. Make sure that if you re-install that
you clean apt first or it will use the one in the cache instead of
downloading a fresh one which just repeats the problem if the downloaded
source is bad.

After completely removing with Synaptic or purging with apt-get issue:
sudo apt-get clean
Then re-install.