Connecting (SD or HD) TV screen to Arduino
I'm just going to post things that I found thanks to comment by Juraj.


"TV Text" library - PAL, SECAM, NTSC 
"VGAX" library - VGA


And with large dedication to everyone who wrote that "it's impossible" I give you a blog post of someone who actually did create he's own NTSC console and a game from scratch (called "The Box") using ATmega328P with external 16Mhz crystal. And he did it in 2013.

http://petenpaja.blogspot.com/2013/11/toorums-quest-ii-retro-video-game.html

Original sources were deleted, but I think they can be found when looking for Toorum Quest 2 

I think I'll try analyzing the VGAX. It looks promising.

EDIT:

There actually is HDMI shield for Arduino!

https://github.com/techtoys/HDMI-Shield/tree/master/Ra8876_Lite

And a nice and very easy VGA tutorial here:

http://www.pyroelectro.com/tutorials/arduino_basic_vga/