The Raspberry Pi (and many similar boards) actually has GPIO's, SPI and I2C that you can control. Not really real-time, but for driving an LED matrix (which I suppose is some kind of display), it may prove more than adequate. Surely it's the fastest way of implementing what you described. The downside is, you may not like having to depend on (and bear the boot times of) a Linux OS. The upside, you'll get access to loads of libraries to implement nice GUIs with HDMI output, you'll get all the required drivers including Ethernet, and a lot of memory (RAM and Flash).
If you really need real-time control and/or more IO's, you could also develop a small MCU-based board that would handle the low-level stuff and communicate with the Pi (or some other SBC) through SPI or USB. Then all the higher-level stuff could be done on SBC's or even PC's, making it versatile.
As you have already been told, you won't find any MCU with an HDMI output. You'll find that on bigger CPU's along with a lot of RAM and an integrated graphics controller. Directly using those on a low-cost, low-volume project is just not doable in my opinion, hence the approach of getting a ready-made module.
You can find a whole range of MCU's with Ethernet controllers though, so there is another option yet, finding an external video module with HDMI output. I remember one from 4D Systems but with VGA output only:
http://www.4dsystems.com.au/product/uVGA_III/ . It's not cheap though, and has limited resolution. Not sure if you can find any with HDMI output. Even if you do, it will probably not be cost-effective at all compared to a solution like a Pi Zero + a custom small MCU board.