Any network of USB devices involves a single host talking to many slave devices. The typical connection to program an Arduino board is, on the USB side:
PC (USB Host) <-> USB Cable <-> Arduino (USB Device)
On the Arduino board you have the serial side of this communication:
USB Serial Adapter <-> TTL serial lines <-> Microcontroller (RX/TX on D0/D1 pins)
The microcontroller itself has no idea that any USB communication is involved. The USB serial adapter functions as a device only, not a host, so plugging it into another USB device (though a cable with device connectors on both ends, rather than the usual host connector on one end) will not do anything.
(There are some Arduino boards, such as the Leonardo and the Micro, that use an ATmega32U4 microcontroller with a USB device interface built in. These work in substantially the same way as above except with the without the serial adapter and TTL serial link. The built-in USB interface also cannot be used as a USB host.)
Arduino USB Host Shield
If you really want to use USB communication, SparkFun sells as USB Host Shield that you can add to a standard Arduino Uno or similar device. The library for it apparently supports USB serial, though you'd want to confirm that before buying the shield. This would leave the Arduino's USB/Serial interface used for programming free to remain hooked up when using the USB host shield, making development easier.
Direct Serial Connection
A better method (IMHO) is to drop the USB stuff altogether and, as Chris Stratton describes in his answer, have the Arduino talk TTL serial directly to a level shifter (e.g., this breakout board or this shield) that is connected to your RS-232 device. I would go with communications on non-programming pins (i.e., not digital pins 0 or 1) using the SoftwareSerial library or the AltSoftSerial library. (You don't mention the speed of the link, but software serial is generally ok up to 57,600 bps and, depending on the device and what else you're doing, even 115,2000 bps.) This will let you use the serial link to the computer for debugging while running your application.
If you need to work at higher speeds or you just prefer to use a UART (hardware serial), you have two options:
Use the USART on digital pins 0 and 1, and be unable to talk to the computer while you're using it to communicate with your RS-232 device.
Use an Arduino board with more UARTs or USARTs on it, such as the Mega.