HDMI (and HD video in general) requires a very large bandwidth, which implies a fast processor that is capable of generating or manipulating that much data.
This excludes microcontrollers. If you want to drive HDMI, the smallest chip that could do so would be a small FPGA, an ARM Cortex A series, or similar. Microcontrollers simply don't have the required clock speed, interconnect bandwidth, or memory. Also, HDMI requires differential signal drivers, which are exceedingly rare if not nonexistent on microcontrollers.
Now, that said, it may be possible to couple a microcontroller to an HDMI transmitter, but you will still be hampered by the slow clock speed and very limited RAM. You will be better off using either an ARM A8/9 or an FPGA (probably with external DDR).
Typically, when you need a microcontroller-attached display, you would use one of the following options:
- a small LCD display, typically 16x2 or 20x4 monochromatic line display. These can be driven directly via GPIO pins, or through a serial/I2C "backpack" which reduces the number of needed IO lines.
- an I2C/SPI OLED. These are often 128x64 pixels, monochromatic.
- a SPI TFT LCD. Full-color, typ. 1.8-~4in diagonal, sometimes with resistive touch support.
It is possible to directly drive VGA at limited resolutions and color depths from a MCU, but it requires the majority of available processing cycles and memory, so it is typically more often a novelty rather than a practical solution.