The military university is grayed out and I can no longer promote troops. How do I fix this?

You probably can't promote troops because:

1. the lower level troops are on a march and can only be promoted once they are back in the castle, doing this will remove those troops in the legion formation

2. you haven't researched the troop promotion for that tier yet, tech tree located at the top row, right-most

3. you haven't upgraded the troop building to the next troop tier