In most nations, the Interior Minister is responsible for police or security. However, in some countries, an Interior Minister can be more in charge of Internal Matters, such as entry into the country (like the Israeli Interior Minister).
The United States has a United States Secretary of the Interior, but that post is more involved in parks and wildlife. A more appropriate similarity would be the United States Secretary of Homeland Security. In the United Kingdom, the equivalent is the Secretary of State for Home Affairs.