¿Es posible conectar mi aplicación de Android a la base de datos mysql sin PHP?

Yes, You can.

Debe agregar una biblioteca jar para mysql. Luego, código java que se conectará al servidor mysql.

import java.sql.DriverManager;

import android.database.SQLException;

import com.mysql.jdbc.Connection;
import com.mysql.jdbc.ResultSet;
import com.mysql.jdbc.Statement;


    public class SQLConnection {
        public static Connection conn = null;
        static void setConnection(String user,String pass,String urlServer,String dBase){   


              try
              {
                  String userName = user;
                  String password = pass;
                  String url = "jdbc:mysql://"+urlServer+"/"+dBase;

                  Class.forName ("com.mysql.jdbc.Driver").newInstance ();
                  conn = DriverManager.getConnection (url, userName, password);
                  System.out.println ("Database connection established");
              }
              catch (Exception e)
              {
                  System.err.println (e.getMessage());
              }
              finally
              {
              }


        }
        public static Connection getConnection(){
        return conn;
        }

    }