I have searched but I did not find any hadeeth that the Prophet (peace and blessings of Alah be upon him) used to recite Surah al-‘Asr before leaving a gathering. Is it an innovation?
Abu Dawood (4859) narrated that Abu Barzah al-Aslami (may Allah be pleased with him) said: The Messenger of Allah (peace and blessings of Allah be upon him) used to say, when he wanted to leave a gathering: “Glory and praise be to You, O Allah, there is no god but You, I seek Your forgiveness and I repent to You.” A man said: O Messenger of Allah, you are saying something that you did not say before. He said: “It is expiation for anything that happened in the gathering.”
It was classed as saheeh (authentic) by al-Albani in Saheeh Abi Dawood.
The knowledge and virtue of the Sahabah (may Allah be pleased with them) and the fact that they so closely followed the Prophet (peace and blessings of Allah be upon him) indicate that whoever among them knew this hadeeth (narration) acted upon it, and we do not need to prove that from each one of them. Rather the basic principle concerning them is that they acted upon the hadeeth and followed the Prophet (peace and blessings of Allah be upon him).
With regard to reciting Surah al-‘Asr before leaving a gathering, there is a report which indicates that this was the practice of the Sahabah (may Allah be pleased with them).
It was narrated that Abu Madeenah al-Darimi said: When two men among the companions of the Prophet (peace and blessings of Allah be upon him) met, they did not part until one of them recited to the other: 1. “By al‑‘Asr (the time). 2. Verily, man is in loss” [i.e., Surah al-‘Asr], then one would say salam to (greet) the other. Narrated by Abu Dawood (no. 417); al-Tabarani in al-Mu’jam al-Awsat (5/215); al-Bayhaqi in Shu’ab al-Eeman (6/501), via Hammad ibn Salamah, from Thabit al-Banani, from Abu Madeenah al-Darimi. Classed as saheeh by al-Albani in al-Silsilah al-Saheehah (2648). He commented on it by saying:
From this hadeeth we learn two things that were among the actions of all the righteous predecessors (may Allah be pleased with them all):
The first is to say salam when parting;
Another thing we learn is that the Sahabah (may Allah be pleased with them) adhered to this practise, namely reciting Surah al-‘Asr, because they are the least likely of any people to introduce into the religion an act of worship by means of which they seek to draw close to Allah, except on the basis of what they learned from the Messenger of Allah (peace and blessings of Allah be upon him) in word or in deed or by his approval. How could it be otherwise, when Allah, may He be blessed and exalted, praised them in the highest terms, saying (interpretation of the meaning): “And the foremost to embrace Islam of the Muhajiroon and the Ansar and also those who followed them exactly (in Faith). Allah is well-pleased with them as they are well-pleased with Him. He has prepared for them Gardens under which rivers flow (Paradise), to dwell therein forever. That is the supreme success” [al-Tawbah 9:100]?
And Allah knows best.