Short answer
Yes. The sound report says Fatima asked Abu Bakr for her share of what her father had left. When he gave her the Prophet's (ﷺ) words, she was angry and did not speak to him before she died. Her family may also have been hurt that they were not consulted about the leadership. But after she died, Ali sought reconciliation with Abu Bakr and gave him his allegiance. We honour her. Abu Bakr gave his reason: the Prophet (ﷺ) said that prophets leave no inheritance, and what they leave is charity. Ali and al-Abbas later confirmed, when Umar asked them by Allah (﷾), that he said it. Abu Bakr said the Prophet's (ﷺ) relatives were dearer to him than his own. No one is blamed here.
Full explanation
What the sound report says
Aisha reports in Sahih al-Bukhari what happened after the Prophet (ﷺ) died. Fatima asked Abu Bakr for her share of what he had left. Abu Bakr told her what the Prophet (ﷺ) had said: our property is not inherited; what we leave is charity. Fatima was angry, and did not speak to him until she died, six months later. We tell it as it is, and with respect.
Aisha reports that after the Prophet's (ﷺ) death Fatima asked Abu Bakr for her share of what the Messenger (ﷺ) had left. Abu Bakr gave her the Prophet's (ﷺ) words: our property is not inherited; whatever we leave is charity. Fatima was angry and did not speak to Abu Bakr until she died, six months later. Abu Bakr said he would not leave anything the Messenger (ﷺ) used to do, for fear of going astray. Later Umar gave the Medina charity to Ali and al-Abbas to manage, and kept Khaybar and Fadak under the ruler.
Aisha reports that Fatima and al-Abbas came to Abu Bakr asking for their share of the Prophet's (ﷺ) property at Fadak and Khaybar. Abu Bakr said he had heard the Messenger of Allah (ﷺ) say that his property cannot be inherited and is to be spent in charity, though the family of Muhammad may take their provisions from it. He added that he would not leave what he had seen the Messenger (ﷺ) do. Fatima left him and did not speak to him until she died.
The leadership may have weighed on them too
The same long report in Sahih al-Bukhari shows a second matter. Ali had not given his pledge to Abu Bakr during those six months. When he did speak to him, he said the family knew Abu Bakr's excellence and did not envy it. But Abu Bakr had not consulted them about the leadership. And they had thought they had a right in it, because of their kinship to the Messenger (ﷺ). So Fatima's displeasure may also have come from this: her husband and family were not asked when the leadership was settled. That is not where the story ends. After Fatima died, Ali himself sought to make peace with Abu Bakr, and gave him his allegiance.
In a long report from Aisha, Ali, who had not given his pledge during the six months of Fatima's life after the Prophet's (ﷺ) death, seeks reconciliation with Abu Bakr once she has died. He sends for Abu Bakr, and Abu Bakr comes to him. Ali says the family know Abu Bakr's excellence and what Allah has given him, and do not envy it, but that Abu Bakr had not consulted them, and they had thought they had a right in the matter because of their kinship to the Messenger (ﷺ). Abu Bakr weeps and says that kindness to the relatives of the Messenger (ﷺ) is dearer to him than kindness to his own. Ali gives him his pledge of allegiance that afternoon and praises Abu Bakr's right in public. The Muslims are glad and draw close to Ali again. The same report begins with Fatima's request about her inheritance; that part is discussed on the page "Was Fatima angry with Abu Bakr?".
We honour both
Fatima was the daughter of the Messenger of Allah (ﷺ) and the mother of al-Hasan and al-Husayn. Muslims honour her, and the Prophet (ﷺ) said she is a part of him. Fatima asked for what she believed was her right, and Abu Bakr answered with what he had heard from her father. We honour them both. Allah (﷾) knows the hearts, and we leave them to Him.
Al-Miswar ibn Makhrama reports that the Messenger of Allah (ﷺ) said: Fatima is a part of me, and whoever makes her angry makes me angry.
Zayd ibn Arqam reports that at a watering place called Khumm, between Mecca and Medina, the Prophet (ﷺ) praised Allah and said he was leaving two weighty things among the people: the Book of Allah, which holds right guidance and light, so hold fast to it; and the people of his household, reminding the people of their duties to the members of his family. Zayd said the Prophet's (ﷺ) wives are of his household, but the household meant here are those to whom charity is forbidden: the family of Ali, of Aqil, of Ja'far and of al-Abbas.
Why Abu Bakr acted as he did
Abu Bakr did not act on his own opinion. He acted on the words of the Prophet (ﷺ). He said he would not leave anything the Messenger of Allah (ﷺ) used to do. He feared going astray if he left any part of it. The Qur'an says to take what the Messenger assigns, and to leave what he withholds. It also names the kindred, orphans and the needy among those who share in what Allah (﷾) bestows on His Messenger. Abu Bakr said that the family of Muhammad may take their provisions from the property.
Aisha reports that after the Prophet's (ﷺ) death Fatima asked Abu Bakr for her share of what the Messenger (ﷺ) had left. Abu Bakr gave her the Prophet's (ﷺ) words: our property is not inherited; whatever we leave is charity. Fatima was angry and did not speak to Abu Bakr until she died, six months later. Abu Bakr said he would not leave anything the Messenger (ﷺ) used to do, for fear of going astray. Later Umar gave the Medina charity to Ali and al-Abbas to manage, and kept Khaybar and Fadak under the ruler.
Aisha reports that Fatima and al-Abbas came to Abu Bakr asking for their share of the Prophet's (ﷺ) property at Fadak and Khaybar. Abu Bakr said he had heard the Messenger of Allah (ﷺ) say that his property cannot be inherited and is to be spent in charity, though the family of Muhammad may take their provisions from it. He added that he would not leave what he had seen the Messenger (ﷺ) do. Fatima left him and did not speak to him until she died.
“What Allah has bestowed on His Messenger (and taken away) from the people of the townships,- belongs to Allah,- to His Messenger and to kindred and orphans, the needy and the wayfarer; In order that it may not (merely) make a circuit between the wealthy among you. So take what the Messenger assigns to you, and deny yourselves that which he withholds from you. And fear Allah; for Allah is strict in Punishment.”
The Prophet's (ﷺ) words, confirmed by Ali and al-Abbas
Years later, Ali and al-Abbas came to Umar disputing over the property. Umar asked them both, by Allah (﷾), if they knew that the Messenger (ﷺ) said: our property is not inherited; what we leave is charity. Ali and al-Abbas answered: he said so. Umar then handed them the charity of Medina to manage. The condition was that they do as the Prophet (ﷺ), Abu Bakr and he had done. They accepted. So the Prophet's (ﷺ) words were confirmed by Ali himself.
Malik ibn Aws reports that Ali and al-Abbas came to Umar disputing over the property of Banu al-Nadir. Umar asked a group, and then Ali and al-Abbas, by Allah, whether they knew that the Messenger of Allah (ﷺ) said: our property is not inherited; what we leave is charity. They answered: he said so. Umar then told them how Abu Bakr and he had managed it as the Prophet (ﷺ) did, and that he handed it to them on that condition, which they accepted.
The Prophet's (ﷺ) relatives were dearer to Abu Bakr
In the long report, Ali says that they acknowledge Abu Bakr's superiority. He speaks of their kinship to the Messenger (ﷺ). Abu Bakr then swears by Allah (﷾) that doing good to the relatives of the Messenger of Allah (ﷺ) is dearer to him than doing good to his own. Abu Bakr also told the people to honour Muhammad (ﷺ) through his family.
In Aisha's report of Fatima's request about her inheritance, Abu Bakr gives the Prophet's (ﷺ) words and says he will not change what the Messenger (ﷺ) did. Ali then says that they acknowledge Abu Bakr's superiority and mentions their own kinship to the Messenger (ﷺ). Abu Bakr answers that doing good to the relatives of the Messenger (ﷺ) is dearer to him than doing good to his own relatives.
In a long report from Aisha, Ali, who had not given his pledge during the six months of Fatima's life after the Prophet's (ﷺ) death, seeks reconciliation with Abu Bakr once she has died. He sends for Abu Bakr, and Abu Bakr comes to him. Ali says the family know Abu Bakr's excellence and what Allah has given him, and do not envy it, but that Abu Bakr had not consulted them, and they had thought they had a right in the matter because of their kinship to the Messenger (ﷺ). Abu Bakr weeps and says that kindness to the relatives of the Messenger (ﷺ) is dearer to him than kindness to his own. Ali gives him his pledge of allegiance that afternoon and praises Abu Bakr's right in public. The Muslims are glad and draw close to Ali again. The same report begins with Fatima's request about her inheritance; that part is discussed on the page "Was Fatima angry with Abu Bakr?".
Abu Bakr said: watch over (honour) Muhammad (ﷺ) through his family.
Ali's pledge
About six months after the Prophet's (ﷺ) death, once Fatima had died, Ali sent for Abu Bakr, and Abu Bakr came to him. Ali spoke well of Abu Bakr's right and gave him his pledge. Abu Bakr wept. The Muslims were glad, and drew close to Ali again. Ali also named Abu Bakr the best of people after the Prophet (ﷺ). Her husband Ali chose the way of peace.
In a long report from Aisha, Ali, who had not given his pledge during the six months of Fatima's life after the Prophet's (ﷺ) death, seeks reconciliation with Abu Bakr once she has died. He sends for Abu Bakr, and Abu Bakr comes to him. Ali says the family know Abu Bakr's excellence and what Allah has given him, and do not envy it, but that Abu Bakr had not consulted them, and they had thought they had a right in the matter because of their kinship to the Messenger (ﷺ). Abu Bakr weeps and says that kindness to the relatives of the Messenger (ﷺ) is dearer to him than kindness to his own. Ali gives him his pledge of allegiance that afternoon and praises Abu Bakr's right in public. The Muslims are glad and draw close to Ali again. The same report begins with Fatima's request about her inheritance; that part is discussed on the page "Was Fatima angry with Abu Bakr?".
Ali's son Ibn al-Hanafiyya asked his father who the best of people were after the Messenger of Allah (ﷺ). Ali answered Abu Bakr, then Umar. Afraid that Ali would name Uthman next, he asked: then you? Ali replied that he was only a man among the Muslims.
Perspectives
Some Muslims believe Fatima's anger shows that Abu Bakr wronged her
Some Muslims, from love of Fatima, hold that her anger was just, because the Prophet (ﷺ) said that whoever angers her angers him. They bring these arguments.
The Prophet (ﷺ) said: Fatima is a part of me, and whoever makes her angry makes me angry.
Rebuttal
We affirm her honour fully, and these words are true. Abu Bakr did not act from his own wish or for his own gain. He followed her father's own words, which Ali and al-Abbas later confirmed. He swore that the Prophet's (ﷺ) relatives were dearer to him than his own.
The Qur'an speaks of prophets leaving heirs, as Solomon was David's heir and Zakariyya asked for an heir.
Rebuttal
Ibn Kathir explains both as the inheritance of prophethood and knowledge, not wealth. He ties this to the Prophet's (ﷺ) words that prophets leave charity.
Fatima asked for what her father had given her, and Abu Bakr refused.
Rebuttal
The sound report says Fatima asked for her share of inheritance. Abu Bakr answered with the Prophet's (ﷺ) words, and he said the family may take their provisions from it.
Ali said Abu Bakr had not consulted the Prophet's (ﷺ) family about the leadership, though they thought they had a right in it. So the family were wronged.
Rebuttal
Ali did say this. But in the same breath he said the family knew Abu Bakr's excellence, and did not envy what Allah (﷾) had given him. After Fatima died, Ali sought to make peace with Abu Bakr and gave him his allegiance in public. The Muslims said he had done the right thing. Ali later named Abu Bakr the best of people after the Prophet (ﷺ).
Ali delayed his pledge to Abu Bakr, which shows he disagreed.
Rebuttal
Ali gave his pledge and praised Abu Bakr's right. He said the family did not envy what Allah (﷾) had given him. The Muslims were glad. And Ali named Abu Bakr the best after the Prophet (ﷺ).
In the fuller report, Umar says that Ali and al-Abbas had thought ill of Abu Bakr. So they never accepted his ruling.
Rebuttal
Umar answers in the same report that Allah (﷾) knows Abu Bakr was truthful and rightly guided. In the same report, Umar asks them both by Allah (﷾), and both confirm the Prophet's (ﷺ) words. And they accept the property on the Prophet's (ﷺ) terms, as Abu Bakr and Umar had kept them.
The Islamic perspective
Fatima was angry, and we honour her; her family may also have been hurt that they were not consulted about the leadership. Abu Bakr followed the Prophet's (ﷺ) words, which Ali and al-Abbas confirmed when Umar asked them by Allah (﷾). After her death Ali sought reconciliation and pledged to him. We honour Fatima and Abu Bakr both, and leave the rest to Allah (﷾).
Fatima asked Abu Bakr for her share. When Abu Bakr told her the Prophet's (ﷺ) words, she was angry with him and did not speak to him before she died.
Abu Bakr acted on the Prophet's (ﷺ) words that prophets leave charity, not inheritance.
Ali and al-Abbas confirmed, when Umar asked them by Allah (﷾), that the Prophet (ﷺ) said it.
Abu Bakr said the Prophet's (ﷺ) relatives were dearer to him than his own.
After Fatima's death, Ali sought reconciliation with Abu Bakr and gave him his pledge of allegiance.
Believers take what the Messenger assigns and leave what he withholds.