I have a program that uses the GTK+ library however I would like to statically link it to see if I can get it to run on linux platforms that don't necessarily have the correct versions of the required shared libraries.
GTK is licensed under the LGPL, so most people just dynamically link with it. If you statically link, your code must also be licensed as GPL or LGPL.
It is possible to do it. However, you must recompile the GTK library using the static link options. Once you have the static library, you can statically link your program with it.