Many people confuse a 'USB to Serial' adapter with a 'USB to RS232' adapter.
They are not the same.
A USB-RS232 adapter will not work.
RS232 defines logic 0 as a positive voltage, logic 1 as a negative voltage.
The chip on your Arduino expects almost the opposite... positive voltage for logic 1, and near zero volts for logic zero.
In addition, the Arduino board needs to be reset before attempting to upload a sketch.
The Arduino application toggles DTR to reset the board. If you only connect TX and RX, you will not be able to program the Arduino.
Upon reset, an Arduino runs a 'BootLoader' program.
The boot loader then looks for a 'magic' value to come down the serial connection.
If there is Magic, the bootloader programs the Arduino FLASH memory with your sketch.
If there is no Magic, the bootloader just times out after a few seconds, and runs whatever application is already present in FLASH.
Your may or may not have damaged your Arduino by using the RS232 adapter. Probably not... but the possibility is there.
Your Arduino came with the proper USB to Serial adapter. Why not just connect the corresponding cable?