Can I change the flour type of my sourdough starter?

Robert Cartaino's answer hits a main point: sourdough cultures are adapted to what they are typically fed. When switching flours, it may take a while for the culture to adjust. The starter may seem a bit sluggish at first.

That said, I would NOT recommend trying to start a wheat culture if you already have a healthy starter that began with rye. There are two main reasons.

First, rye is particularly rich in certain sugars and food that natural yeasts tend to like, so rye starters naturally tend to look more "lively" than wheat starters, especially in the early stages of a starter. Part of what you may be seeing in your transition to wheat is simply less activity produced because the food source is less rich in nutrients, not necessarily because the yeast and bacteria can't digest it.

Second, a lot of recipes to generate new starters recommend using rye at the beginning, regardless of what you ultimately plan to feed the starter. Because it provides a strong food source for young cultures, it's usually a lot easier to begin with rye (perhaps this is why you did). In my experience, it tends to take new wheat starters at least twice as long to get strongly established compared to rye, and there's a higher chance of failure if you don't use a good recipe.

For both of these reasons, some people even recommend always feeding your starter with rye flour, even if you tend to make wheat-based bread. Since many people have success doing that (and I have personally), I don't think you need to begin a whole new culture with wheat alone.

If your starter really is sluggish (and not just a little less active than it was with rye), you can help your starter to adapt to the new food. I'm not sure what your typical feeding routine is or what the hydration level of your starter is. But the general thing to do in these circumstances is to dilute your starter and feed it large quantities of the new food. For example, if you maintain 200 grams of your starter normally, I would discard all but maybe 40-50 grams and then add enough new flour and water to get back to 200 grams in whatever your usual proportion is. (Some people would be even more extreme in diluting the starter and cut the starter proportion even further.)

Within a few days of feedings like this, you will have strongly selected the yeast and bacteria that thrive on whole wheat and your starter should be strong again. After a week of feedings like this, I doubt you could distinguish the performance of your starter from one that was begun on wheat. (Note that the specific microorganisms may still be slightly different, but your starter should do a fine job raising dough and providing flavor.)

Different flours contain different varieties of naturally-occurring yeasts and bacteria. When you create a sourdough culture, these yeasts and bacteria compete and eventually stabilize to balance out into a favorable equilibrium. Natural selection. Unfortunately, when you changed the type of flour you fed to that culture, you dramatically changed the environment the yeasts optimized for.

It's entirely possible that your rye-based culture simply will not do as well in the wheat flour. Unfortunately, since that culture is already dominant in your starter, it may be competing with the naturally-occurring organisms present in the new flour.

It may balance itself out eventually, but it may simply be better to start a new culture with the type of flour you wish to use in the longer term.