GNU ARM Eclipse - flash and run without debug
Finally I found the solution myself:


Disable semihosting - it seems when the firmware is loaded via OpenOCD,
MCU detects that semihosting interface is available and after disconnecting
of the debugger the MCU hangs when it tries to write the semihosting
interface.

To make firmware running again it is necessary to replug the ST-link USB
connector or leave the debugger running all the time.

I am not experienced in semihosting too much, therefore I just removed all
semihosting debug messages to get rid of it :)
In Eclipse go to Run -> Run Configurations...

a) Make a copy of your Debug configuration.

b) On the Debugger tab update Config options of OpenOCD like this:

-f board/stm32f429discovery.cfg -c "program Debug/blinky.elf verify reset exit"


(of course replace the board definition by your board and also
 replace the Debug/blinky.elf by your binary file name).

c) On Startup page uncheck all the options to leave GDB alone.

d) Save.


After doing the steps above, Eclipse will only program the board without running debugger.