According to the U.S. Department of State website, resident aliens cannot get U.S. passports because one of the requirements of obtaining a passport is proof of U.S. citizenship. Resident aliens are not classified as citizens.
The U.S. Citizenship and Immigration Services points out that resident aliens traveling outside the United States must use either a valid passport from their country of citizenship or a refugee travel document. They cannot obtain a passport unless they first apply for and receive citizenship. To prove citizenship for a passport, applicants must provide a previously issued valid U.S. passport, a certified U.S. birth certificate, a Consular Report of Birth Abroad, a naturalization certificate or a certificate of citizenship.