Can I create a company called Rush Fitness Center if there is already a company called The Rush Fitness Complex since there name is not trademarked?

Why?  Even if it is legal, the similarities in the name is apt to cause confusion among customers and among vendors (presuming your businesses are geographically proximate). Isn't there a name you can choose that is less similar to another company? Even if the company name is not formally trademarked, the name can be a common law trademark. A common law trademark is protectable in the region within which it used to the extent it is recognized by the market. If you go to wikipedia, you can search terms like "common law trademark" and "passing off" for a description of the relevant issues. If you are in SC and The Rush Fitness Complex is limited to Texas, you will likely not have any issues adopting your name for use in SC. The key in trademark law is always consumer confusion. If Rush Fitness Complex customers are likely to see your business and believe your companies are related (thus being confused about a relationship between the two of you), you could have an issue. I always tell my clients to bear in mind, even if the law permits you to do something, a company that feels wronged may sue you anyway. What will it do to your business, just starting out, if you have to pay an attorney five figures to defend your right to use this name? You can survive walking down the middle of the road, but you get where you are going safer if you use the sidewalk.