Would an Executive Order Banning Discrimination on Basis of Sexual Orientation Be Constitutional?

Gay rights groups have strongly criticized President Obama for refusing to issue an Executive Order banning employment discrimination on the basis of sexual orientation by federal contractors. In my opinion, the President could issue an order prohibiting government agencies from discriminating on the basis of sexual orientation. However, I believe that the President lacks the authority to prohibit discriminatory action by private businesses that are under contract with the government.