His wudoo’ was invalidated after saying the tasleem to his right; is his prayer valid or must he repeat it?

The majority of scholars are of the view that the second
tasleem in the prayer is mustahabb (encouraged). The Hanbalis are of the
view that it is obligatory in an obligatory prayer. It says in al-Mawsoo‘ah
al-Fiqhiyyah (11/314): 

The first tasleem to exit from the prayer whilst sitting is
obligatory according to the Maalikis, Shaafa‘is and Hanbalis. The Hanbalis
also regard the second tasleem as obligatory, except in the case of the
funeral prayer and naafil (supererogatory) prayers, because the final part
of the sitting in which the salaam occurs is obligatory. End quote. 

More than one of the scholars narrated that there was
scholarly consensus that the second tasleem is not obligatory. 

Al-Qurtubi (may Allah have mercy on him) said: 

There was no difference of opinion, among those scholars who
say that the tasleem is obligatory and those who say that it is not
obligatory, concerning the fact that the second tasleem is not obligatory,
apart from the report from al-Hasan ibn Hayy which says that he regarded
both tasleems as obligatory. Abu Ja‘far at-Tahhaawi said: we did not find
anyone among the scholars who is of the view that the second tasleem is
obligatory, apart from him.

End quote from Tafseer al-Qurtubi (1/362) 

An-Nawawi (may Allah have mercy on him) said: 

The prominent scholars are unanimously agreed that only one
tasleem is obligatory, but if a person says only one tasleem, it is
mustahabb for him to say it to the front, but if he says two tasleems, then
he should say the first to his right and the second to his left, turning in
each case until his cheek can be seen by the one who is sitting beside him.
This is the correct view.

End quote from Sharh Saheeh Muslim (5/83) 

Ibn Rajab (may Allah have mercy on him) said:

The majority of those who say that there should be two
tasleems are of the view that if the worshipper limits it to one tasleem,
that is sufficient and his prayer is valid. Ibn al-Mundhir stated that there
was consensus on this point among those scholars from whom he acquired
knowledge.

End quote from Fath al-Baari by Ibn Rajab (5/213) 

Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) said: 

If a person limits it to one tasleem, is that sufficient? The
answer is: This is also a matter concerning which the scholars differed.
Some of them said that it is sufficient and others said that it is not. Some
of the scholars said that one is sufficient in the case of a naafil prayer
but not in the case of an obligatory prayer. 

So there are three opinions, but to be on the safe side one
should say two tasleems, because if a person says two tasleems, not one of
the scholars will say that his prayer is invalid, but if he says only one
tasleem, then some of the scholars will say that his prayer is invalid, and
it is well-known that the Prophet (blessings and peace of Allah be upon him)
instructed us to err on the side of caution with regard to matters in which
the evidence is not clear. 

End quote from ash-Sharh al-Mumti‘ (3/211-212) 

Based on that, if a person said one tasleem then his wudoo’
was invalidated, his prayer is valid and he does not have to repeat it,
according to the more correct scholarly opinion, based on what is mentioned
above. 

But if his wudoo’ was invalidated before completing the
tasleem on his right by saying “as-salaamu ‘alaykum”, then his prayer is
invalid and he has to repeat it, because his wudoo’ became invalidated
during one of the obligatory acts of the prayer, so the prayer is no longer
valid. 

An-Nawawi (may Allah have mercy on him) said: 

The tasleem is one of the pillars or essential parts of the
prayer without which it is not valid and nothing else can take its place.
The minimum that is to be said is “as-salaamu ‘alaykum”; if even a single
letter is omitted from this phrase, his tasleem is not valid.

End quote from al-Majmoo‘ (3/475) 

For more information, please see the answers to questions no:
154587 and 105297

And Allah knows best.