Stress activates your autonomic nervous system specifically your sympathetic nervous system. This is commonly known as your fight or flight mechanism. When you go into fight or flight your body increases acid production in order to protect your stomach. Problem is this is a great solution if we're being attacked by bacteria in our food. Not so much when we're being attacked by the stack of papers on our desks.
It's easy to say we need to calm down and take it easier. Sure that would be good for us all. For me, I've found that exercise does indeed help me feel better both physically and mentally. Half an hour a day makes a huge difference. But, my stomach is still a mess either way :)