I am not a medical doctor, but as an engineer and applying common sense, I will venture to say "NO".

When you rinse your mouth after brushing teeth you do not shallow the water (I hope), but spit it out.  So, technically no water will get to your stomach to cause acids there to be present to aid in digestion.  Ulcers are produced when there is an excess of acid present for long periods of time.

In my opinion, only a germ infection, maybe a diarrhea case may develop if you use tap water to rinse your mouth after brushing teeth.  And it would also depend where (which city) you are at.

I use one of those electric pots to boil water, then after it cools down I pour it into a plastic bottle and keep that at the bathroom for rinsing my mouth as needed.  And no problem so far.  I also do boil tap water locally, and afterwards filter it, to be used as drinking water sometimes.  But here fruit juices are so cheap, I do drink lots of those.

Simple answer is no.  There is a chance that if you have really bad tap water, it could futher irritate an existing ulcer in your mouth.  But the little water used to brush your teeth would be near impossible to cause an ulcer.

Though I am not an M.D.  I do not think that tapwater causes ulcers, not even here in China. That level of acidity would have completely destroyed the pipes, tubes, etc.
What may be happening is that it could be your diet. There are some foods that if I eat them too often I get mouth ulcers. Brushing only makes it hurt more, but I cannot live without brushing.
Hope you get better soon!

Double Post...

Hang on, do you mean like mouth ulcers?
I totally notice that when i maybe damage my gums with my toothbrush and gete a small wound or ulcer it takes alot longer to heal naturally here than in my own country. Maybe its just my imagination but now when i do get these i use some iodine mouthwash to disinfect it and they heal pretty quick