Reciting the surah Fatiha at the time of marriage or singling any part of the quran and associate it with some occasion is considered to be an innovation(biddah). There is no basis for this in sunnah. Reciting surah fatiha and making it part of the ceremony has nothing to do with the sunnah of Prophet Muhammad(peace be upon him) neither did any if his companions do that.
It is not permissible to single out any part of the Qur’an for any actions except on the basis of evidence.
Abu Shaamah al-Maqdisi said in al-Baa‘ith ‘ala Inkaar al-Bida‘ wa’l-Hawaadith (165):
It is not appropriate to single out times for acts of worship that were not singled out for such acts in Islamic teachings. People have no authority decide such matters; rather that is the prerogative of the Lawgiver.
Reciting surah fatiha and thinking that it is part of the sunnah is an innovation and is not permissible to follow under the correct rulings of Islam.
Also there is no basis for such tradition as mentioned in the rulings of the marriage contract. Reciting al-Faatihah does not mean that the marriage contract has been completed; rather what matters is the proposal and acceptance with the wali (woman’s guardian) and witnesses.
The Sunnah is to recite khutbat al-haajah at the time of the marriage contract.
It was narrated that ‘Abdullah ibn Mas‘ood (may Allaah be pleased with him) said:
The Messenger of Allah (peace and blessings of Allah be upon him) taught us Khutbat al-Haajah, (to be said) at weddings and on other occasions:
“Inna al-hamda Lillaahi nasta’eenahu wa nastaghfiruhu, wa na’oodhu bihi min shuroori anfusinaa. Man yahdih Illaahu falaa mudilla lahu wa man yudlil falaa haadiya lahu. Wa ashhadu an laa ilaaha ill-Allaah wa ashhadu anna Muhammadan ‘abduhu wa rasooluhu
(Verily, all praise is to Allah, we seek His help and His forgiveness. We seek refuge with Him from the evil of our own souls. Whomsoever Allah guides will never be led astray, and whomsoever Allah leaves astray, no one can guide. I bear witness that there is no god but Allah, and I bear witness that Muhammad is His slave and Messenger).
‘Yaa ayyuha’n-naas uttaqu rabbakum alladhi khalaqakum min nafsin waahidatin wa khalaqa minhaa zawjahaa wa baththa minhumaa rijaalan katheeran wa nisaa’an wa’ttaqu-Llaah alladhi tasaa’aloona bihi wa’l-arhaama inna Allaaha kaana ‘alaykum raqeeban
(O mankind! Be dutiful to your Lord, Who created you from a single person, and from him He created his wife, and from them both He created many men and women, and fear Allah through Whom you demand your mutual (rights), and (do not cut the relations of) the wombs (kinship) Surely, Allah is Ever an All-Watcher over you).’ [al-Nisaa’ 4:1]
‘Yaa ayyuha’lladheena aamanu-ttaqu’Llaaha haqqa tuqaatihi wa laa tamootunna illaa wa antum muslimoon
(O you who believe! Fear Allah as He should be feared, and die not except in a state of Islam (as Muslims) with complete submission to Allah.)’ [Aal ‘Imraan 3:102]
‘Yaa ayyahu’lladheena aamanu-ttaqu’Llaaha wa qooloo qawlan sadeedan yuslih lakum a’maalakum wa yaghfir lakum dhunoobakum wa man yuti’ Allaaha wa rasoolahu fa qad faaza fawzan ‘azeeman
(O you who believe! Keep your duty to Allah and fear Him, and speak (always) the truth). He will direct you to do righteous good deeds and will forgive you your sins. And whosoever obeys Allah and His Messenger, he has indeed achieved a great achievement (i.e. he will be saved from the Hell‑fire and will be admitted to Paradise)’ [al-Ahzaab 33:70, 71].”
Narrated by Abu Dawood (2118) and classed as saheeh by al-Albaani in Saheeh Abi Dawood.
And Allah knows best.