The short answer to your question is that the extra digestible fiber and denser construction of whole-wheat bread means that your stomach has to work harder to break the food down. This turns into more of a roiling motion that can splash acid around and cause heartburn. So the basic answer is 'yes'.
However, you should absolutely not go back to eating white bread. This is a sign from your body that your digestion is out of whack for whatever reason, and you should try to address it directly instead of reverting back to the bad habit of consuming bleached white flour, which is probably what got you into this regrettable scenario to begin with.
I would recommend you look into 'food matching', a concept I touched on in this question, and increase your intake of dark green vegetables to give your body more indigestible fiber to work with. 5 servings a day would be a good place to start if you're not there already. If you do those two things I would not be surprised at all if your problems vanished quickly.
As a final note, usually when diets talk about increasing fiber intake, they mean indigestible fiber that comes from fruits and vegetables. Digestible fiber from starches and grains are the other kind, and generally speaking most people get plenty of that already.