Sheikh Muhammad b. Sâlih al-`Uthaymîn
Passing gas nullifies a persons state of ritual purity and therefore requires the person to perform ritual ablutions (wudû').
The Prophet (peace be upon him) said: "Do not break off from your prayer unless you hear or detect the passage of gas." [Sahîh al-Bukhârî (137) and Sahîh Muslim (361)]
The person has to perform a fresh wudû'; however, he does not have to wash his private area. This is because nothing has come out that requires being washed off.
Cleaning the private parts is not part of wudû’.
Some people are confused about this. They go and wash their private parts whenever they wish to perform wudû', even though they already washed themselves earlier when they went to the bathroom. They wash their private parts all over again unnecessarily because they think that washing the private parts is part of wudû'.
This is wrong. Cleaning the private areas, by either washing them or wiping them off, is only required at the time when a person urinates or defecates – and that is simply for the purpose of cleaning off the residual impurities. Once the private area is clean, then there is nothing else to worry about. It has nothing to do with whether or not the person plans to make wudû’ at that time.
Source: Islam Today