Install an external plugin from within the current plugin?

This solution should do the job for you.

http://tgmpluginactivation.com/

The solution is meant for theme developers to package to download, automatically install and activate multiple plugins that are either packaged with a theme, downloaded from the WordPress Plugin Repository or downloaded elsewhere on the internet (perhaps a private repository.

The code of the solution can be used to accomplish what you are trying to achieve. The code is also available on GitHub at:

https://github.com/TGMPA/TGM-Plugin-Activation

Best of luck.

tech