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Is there any way to change white background when displaying settings, messages, contacts, etc. without using/installing any third party apps?
Changing the theme only changes color of the fonts, as I have noticed. I can't seem to find it under settings, and I don't like the flashing white background.
Thanks!
Thank you for your answers.
@Phy
Unfortunately, I can't find the Color inversion option under Acessability settings. Guess we're stuck with white background. 
Nice. Good to know. I'll change it as soon as I upgrade android version on the phone.
Thanks!
Some Xperia themes change the color/pattern or App Drawer and Settings menu. Give them a try, go to Settings -> Personalization -> Themes -> Try "Get more themes" option which will redirect you to Play Store and choose from a plethora of free and paid themes.
After doing some reading I've found out that this is a Kitkat related issue. In order to be able to change system background color, the phone needs to be rooted. So, no launchers or third party apps can solve this issue.
@Phy
From what I've read, Lollipop upgrade for Xperia E4 is rumored to be out in august.
@Quester
Thanks for the suggestion. I've already tried out couple of themes, but unfortunately none of them can change the system background color. By installing some of the themes, they were able to change wallpaper, lock screen wallpaper and font color. I did see that some of the themes (screenshots) are able to make some sort of an overlay picture in the settings menu, over the white background, but I haven't tried it.
Once again, thank you all for your suggestions.
This theme can be used on Kitkat for example and it changes pretty lot of the settings menu.