There is no report from the Prophet (blessings and peace of Allah be upon him) about reciting Soorat al-Faatihah when offering supplication, although there are so many reports about how he offered supplication and the words of his supplications, but he (blessings and peace of Allah be upon him) did not teach us to do that. This indicates that aiming to do this is a kind of innovation.
The Standing Committee for Issuing Fatwas was asked: Did the Prophet (blessings and peace of Allah be upon him) recite al-Faatihah after offering supplication?
They replied: There is no proof that the Prophet (blessings and peace of Allah be upon him) used to recite al-Faatihah after offering supplication, as far as we know. Hence reciting it after offering supplication is an innovation. And Allah is the source of strength.
End quote from Fataawa al-Lajnah ad-Daa’imah (vol. 1, 2/528)
Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) said: Reciting al-Faatihah before offering supplication or at the end of a supplication is an innovation, because there is no report from the Prophet (blessings and peace of Allah be upon him) to suggest that he used to start his supplication by reciting al-Faatihah or that he used to end his supplication by reciting it. If an act of worship was not narrated from the Prophet (blessings and peace of Allah be upon him), introducing it comes under the heading of innovation.
End quote from Majmoo‘ Fataawa Ibn ‘Uthaymeen (14/159)
Shaykh ‘Abd ar-Rahmaan ibn Naasir al-Barraak (may Allah preserve him) said: Reciting al-Faatihah when concluding supplication is an innovation for which there is no basis in the Qur’an or Sunnah, or the actions of the Sahaabah and those who followed them in truth. Hence it is not permissible to do that. Singling out a dhikr or recitation for a particular time, situation or place is not permissible except on the basis of evidence. Allah has enjoined recitation of al-Faatihah in the prayer, and it is a ruqyah that may be recited over one who is sick, and reciting it is an act of worship like any other soorah of the Qur’an. Indeed it is the best soorah of the Qur’an, but I do not remember any particular situation in which it is prescribed to recite al-Faatihah, apart from what is mentioned above. End quote.
And Allah knows best.