'Umar said: I shall not appoint a successor, for the Messenger of Allah ﷺ did not appoint a successor. If I appoint a successor (I can do so), for Abu Bakr had appointed a successor. He Ibn 'Umar) said: I swear by Allah, he did not mention (anyone) but the Messenger of Allah ﷺ and Abu Bakr. So I learnt he would not equate anyone with the Messenger of Allah ﷺ, for he did not appoint any successor.

عُمَرُ إِنِّي إِنْ لاَ أَسْتَخْلِفْ فَإِنَّ رَسُولَ اللَّهِ ﷺ لَمْ يَسْتَخْلِفْ وَإِنْ أَسْتَخْلِفْ فَإِنَّ أَبَا بَكْرٍ قَدِ اسْتَخْلَفَ قَالَ فَوَاللَّهِ مَا هُوَ إِلاَّ أَنْ ذَكَرَ رَسُولَ اللَّهِ ﷺ وَأَبَا بَكْرٍ فَعَلِمْتُ أَنَّهُ لاَ يَعْدِلُ بِرَسُولِ اللَّهِ ﷺ أَحَدًا وَأَنَّهُ غَيْرُ مُسْتَخْلِفٍ

I heard the people saying something so I decided that I should talk to you. They are saying that you are not going to appoint a successor. He lowered his head for a while, then he looked up and said: Allah, may He be glorified and exalted, will protect His religion. If I do not appoint a successor, the Messenger of Allah ﷺ did not appoint a successor either; if I do appoint a successor, Abu Bakr appointed a successor. By Allah, once he mentioned the Messenger of Allah ﷺ and Abu Bakr, I realized that he was not going to regard anyone else as equal to the Messenger of Allah ﷺ and that he was not going to appoint a successor.

أَنْ أَقُولَهَا لَكَ زَعَمُوا أَنَّكَ غَيْرُ مُسْتَخْلِفٍ فَوَضَعَ رَأْسَهُ سَاعَةً ثُمَّ رَفَعَهُ فَقَالَ إِنَّ اللهَ ﷻ يَحْفَظُ دِينَهُ وَإِنِّي إِنْ لَا أَسْتَخْلِفْ فَإِنَّ رَسُولَ اللهِ ﷺ لَمْ يَسْتَخْلِفْ وَإِنْ أَسْتَخْلِفْ فَإِنَّ أَبَا بَكْرٍ قَدِ اسْتَخْلَفَ قَالَ فَوَاللهِ مَا هُوَ إِلَّا أَنْ ذَكَرَ رَسُولَ اللهِ ﷺ وَأَبَا بَكْرٍ فَعَلِمْتُ أَنَّهُ لَمْ يَكُنْ يَعْدِلُ