What must be done in wudoo’ is washing the feet; it is not sufficient to wipe them. Your friend’s understanding of the verse as meaning that the feet may be wiped is not correct.
The evidence that it is obligatory to wash the feet is the report narrated by al-Bukhaari (163) and Muslim (241) from ‘Abd-Allaah ibn ‘Amr (may Allaah be pleased with him) who said: The Prophet (peace and blessings of Allaah be upon him) stayed behind us on a journey and then caught up with us, and we were late in praying ‘Asr. We started doing wudoo’ and wiping out feet, and he called out at the top of his voice: “Woe to the heels from the fire” two or three times.
Muslim (242) narrated from Abu Hurayrah (may Allaah be pleased with him) that the Prophet (peace and blessings of Allaah be upon him) saw a man who had not washed his heels and he said, “Woe to the heels from the Fire.”
Ibn Khuzaymah said: If wiping were sufficient to discharge the obligation, there would have been no warning of the Fire in this case.
Al-Haafiz Ibn Hajar said:
There are mutawaatir reports from the Prophet (peace and blessings of Allaah be upon him) which describe his wudoo’ and state that he washed his feet. He is the one who explained the commands of Allaah. There is no report from anyone among the Sahaabah which differs from that, except from ‘Ali, Ibn ‘Abbaas and Anas, but it was narrated that they retracted that. ‘Abd al-Rahmaan ibn Abi Layla said: The companions of the Messenger of Allaah (peace and blessings of Allaah be upon him) are unanimously agreed that the feet should be washed. Narrated by Sa’eed ibn Mansoor. End quote.
Fath al-Baari, 1/320
With regard to the verse, it says (interpretation of the meaning):
“O you who believe! When you intend to offer As-Salaah (the prayer), wash your faces and your hands (forearms) up to the elbows, rub (by passing wet hands over) your heads, and (wash) your feet up to the ankles”
This does not indicate that it is permissible to wipe the feet. The reason for this is that there are two readings of this verse.
1 – Wa arjulakum (and your feet), with a fathah on the laam. In this case the word “feet” is mentioned in conjunction with the word wajh (face), and the face is to be washed, so the feet are to be washed too. So it is as if the verse is basically saying: “Wash your faces, your arms up to the elbows and your feet up to the ankles, and wipe your heads,” but mention of washing the feet is put after mention of wiping the head so as to indicate that this is the order in which the parts of the body are washed in wudoo’: washing the face, then the arms, then wiping the head, then washing the feet.
See al-Majmoo’, 1/471
2 – Wa arjulikum, with a kasrah on the laam. In this case it is mentioned in conjunction with the word ra’s (head), and the head is to be wiped, so the feet are to be wiped too.
But the Sunnah shows that one may wipe over the feet only when wearing leather slippers or socks, subject to the conditions that are well known in the Sunnah.
See al-Majmoo’, 1/450; al-Ikhtiyaaraat, p. 13
For more information on the conditions for wiping over the socks, please see question no. 9640.
Thus it is clear that in neither reading does the verse indicate that the feet may be wiped. Rather it indicates that it is obligatory to wash the feet, or to wipe over the socks if one is wearing socks.
Some of the scholars, based on the second reading, are of the view that the reason why wiping is mentioned with regard to the feet although they are to be washed is to indicate that one should be economical in using water when washing the feet, because people are usually lavish with water when washing them. So the verse enjoins wiping in the sense of washing them without being extravagant in the use of water.
Ibn Qudaamah said in al-Mughni, 1/186:
It may be that what is meant by wiping is washing lightly. Abu ‘Ali al-Faarisi said: The Arabs call a light washing mash (wiping) and say tamassahtu li’l-salaah (literally “I wiped myself for prayer”) meaning I did wudoo’. End quote.
Ibn Taymiyah said:
Mentioning wiping with regard to the feet is an indication that one should not use too much water on the feet, because people are usually extravagant in using water for that. End quote.
Manhaaj al-Sunnah, 4/174
And Allaah knows best.