Arduino Uno to send commands to Syringe Pump

I am new to electronics and am hoping to use an Arduino Uno to send ASCII commands to a syringe pump. After downloading an FTDI chip driver I can successfully send commands to the pump via USA cable from my PC using a Serial Port Terminal. I built a breadboard circuit and wrote the code to send commands to the Arduino Serial Monitor PC when I press a push button. Now that I can do each separately my question is how to send the commands to the pump instead of the PC. I matched the baud rates, parity, etc necessary for the pump and have tried connecting the pump's USB cable wires to the GND, Rx and Tx of the Arduino. Do I need a separate USB Host Shield even though I am just sending commands to the pump using the Rx Tx pins and Software Serial?

I matched the baud rates, parity, etc necessary for the pump and have tried connecting the pump's USB cable wires to the GND, Rx and Tx of the Arduino

You need to be connecting the device directly to the USB port, so the data goes through the USB-to-serial converter.

Why use the RX and TX pins and SoftwareSerial? Those pins are tied to the Serial instance.

I tried that as well (using a Type 2 USB Male to Type 2 USB Male cable) and directly connecting the two. I also tried splicing two USB cables and connecting directly to the USB ports but with the TX and RX pin swapped

The syringe pump must then be a USB slave, the same as the Arduino. Two slaves can't make each other do anything. You will need a USB Host (master) shield in between.

Yes, that was what I was not sure about. If I use a USB Host shield to make the Arduino the master, will I still be able to use the Serial library and communicate with the device the same as I do the PC? Or do I need the same type of FTDI chip driver that I installed in the PC?