You may be able to do this with extra care taken to watch out for the differences between the languages. The C++ objects can only be accessed within a C++ source file. Anything that needs to be accessed by your C code needs to be made accessible using the keyword :-
This stops the name mangling that would be done normally in C++. It also stops function and name over-riding and other C++ features relating to the variable or function.
If you are not going to be using gtkmm then it may be better to stay with C, especially since the rest of the project is already in C.
Note you do not have to use gcc because you are using GTK+. Code using the GTK+ libraries can be compiled using other compilers.