Update: This is a core feature since J! 3.7.3 release.
We can now sort articles by Date Modified, using the Sorting Dropdown on the top-right side of the articles manager list.
I worked on this feature for a customer and have submitted the code to be considered for inclusion with a future Joomla release.
The patch has now been merged and seems it is in the milestones for the next 3.7.3 version.
When it becomes available, it will be possible to select the Articles Sorting Order by Date Modified and the corresponding Date Column will turn to Date Modified.
If anyone is in hurry for this feature, he can go and get the patch from here: