RTS (request to send) is from transmitter to receiver, and it has no direct effect on the UART core in the receiver. The software can read it, and maybe an interrupt can be generated when it changes state.
CTS (clear to send) is more important. The receiver sets it when it can receive characters. It is normally wired so the "TX register empty" bit is blocked in the transmitter when CTS is not set. That will cause a pause in the transmitter.
Some stupid UART's stopped the transmission in the middle of a character if CTS went inactive, but that is not what you want.