The damage done to your gut by the immune system reaction to gluten definitely can and often does make people sensitive to other foods, but often only temporarily. If a person actually does have celiac disease they must absolutely avoid even the slightest trace of gluten for the rest of their life, no ifs, ands, or buts.
Sensitivities to other foods in a person just starting a gluten free diet are very common. The most common problem is lactose intolerance, which usually does go away after 6 months or a year. Try avoiding all dairy products for two weeks or a month. Dairy foods are a poor choice anyway despite their excellent marketing campaigns. Cows have four stomachs, humans do not.
If I had to guess, based on what you report eating, I'd say that dairy products should be the next thing you try to eliminate.
Generally speaking food intolerance problems tend to be caused by proteins, so during a recovery from gastric problems a diet of white rice, fruit, and well cooked vegetables can often help.
Most people tolerate potatoes pretty well. Soy products often are a problem for people with an irritable gut. Other types of beans may be easier to tolerate.
I hope this helps!