No,it's not possible at all. A template that is not responsive can't be turned into a responsive one.
Thus if you want to use the same desktop template with Responsivizer obviously nothing will change.
The easiest approach that you can find to have a mobile and responsive site is the one offered by Responsivizer. Indeed your desktop template will be replaced with a mobile optimized one only when mobile devices are detected.
The optimized template framework that is embed into Responsivizer will be used when mobile devices are detected, a set of tools will make it possible transition your desktop look and feel also to mobile layout in few easy steps.
You can use Responsivizer with a custom mobile template different than the default one, however take care that in this case you would lose a lot of advanced features only available when using the default template framework.
If you need to fix only certain pages of your desktop template that are not mobile friendly, you can consider to activate the Responsivizer template only on certain areas of the website excluding pages already mobile friendly.