Does touching the private part invalidate wudoo’?

Touching the private part without a barrier invalidates
wudoo’ according to many scholars among the Sahaabah and those who came
after them of the Taabi‘een and the imams, including Maalik, al-Shaafa‘i
and Ahmad. They quoted as evidence for that several hadeeths, including
the words of the Prophet (blessings and peace of Allah be upon him):
“Whoever touches his private part, let him do wudoo’.” Narrated by Abu
Dawood (181) and classed as saheeh by al-Albaani in Saheeh Abi Dawood. 

Others were of the view that touching the private part does
not invalidate wudoo’, as is the view of Abu Haneefah. 

Some of the scholars differentiated between touching the
private part with desire and without desire. Wudoo’ is invalidated if it is
touched with desire, and it is not invalidated if it is without desire. 

This is a very strong opinion and was regarded as such by
Shaykh Ibn ‘Uthaymeen in al-Sharh al-Mumti‘. He stated clearly that
he regarded it as more correct in his commentary on Buloogh al-Maraam. 

To sum up: if a person touches his private part, it is
mustahabb for him to do wudoo’ in all cases, whether he touched it with
desire or not. If he touched it with desire, then the view that it is
obligatory (for him to do wudoo’) is very strong. 

End quote from al-Sharh al-Mumti‘, 1/234 

With regard to touching it from behind a barrier, that does
not invalidate wudoo’. 

Al-Mardaawi said in al-Insaaf (1/202): The apparent
meaning of his words, “touching the private part with his hand”, is that the
touching occurs without a barrier, and this is the correct view. 

Ibn ‘Uthaymeen said in al-Sharh al-Mumti‘ (1/228):
Because if there is a barrier, it is not regarded as touching. 

And Allah knows best.