I think the short answer is no but maybe someone has a set up for this.
What you are asking for is a GUI which isn't a GUI so I don't think that is possible. I have used both Vim and Emacs (Emacs now currently) and do all my editing in the terminal.
Here is my set up which may add what you need:
I use a slightly see through terminal that drops down from the top with the pdf underneath it. This way I can see through it or just hit F12 and it auto hides until I want it back. Then if I want to update the pdf, I just compile the file again and the updates appear on the pdf immediately. If you use synctex, you can then click on the pdf and have it jump to that spot in your terminal window for correcting or editing. In my view, this is the best you may get since you aren't using the GUI.
As gvim is just vim “in gtk-window”, I will use vim for convenience.
When you installed vim via the packagemanager, it should have gotten its own directory under /etc with (at least one) template vimrc.
/etc/vim/vimrc
/etc is usually the place, where configuration-files are stored that apply to the whole system and will be used as fallback, when a user has not defined his own.
Your personalized configuration-file should reside below your homedirectory.
So, usually, it suffices to just have /home/my_home/vimrc.
It is always a good idea to copy the configuration-file from /etc/vim and change that as needed.
If you so far used vim without a personalized vimrc, then it loaded the /etc/vim/vimrc and you might notice a different behaviour when you start with an empty vimrc.
I would recommend to use the hidden directory for vim, in which you can store all vim-related configurations,
i.e. /home/my_home/.vim/vimrc
and plugins, such as the latex-suite.
As for the latex-plugin:
When decompressing the archive, you might only get the following directory:
~/.vim/vim-latex-1.8.23
But what you need, is to have that directories contents to be directly under ~/.vim.
For example:
~/.vim/vim-latex-1.8.23/latextags
→
~/.vim/latextags
So, the plugin (and its documentation) expects a certain order in the directory-tree, without which, the plugin will not be able to function.
I encountered the same problem at first and finally found this solution.
I hope that this can still be of some help.
Best Answer
Thanks to Dai Bowen in the comments, I was able to find the problem. It seems like the Vim-Latex plugin only loads once you open an already existing .tex file, not when one is created in Vim.
I simply created a new file and saved it as a .tex file and then reopened it.