He divorced his wife on the basis of a false accusation

If you only divorced your wife on this basis, then you realized that she was innocent of that, then the divorce  does not count, because it was based on a reason, then it became apparent that this reason had no substance. This is the view favoured by Shaykh al-Islam Ibn Taymiyah and Ibn Rajab, and it has been stated in fatwas issued by two contemporary shaykhs: Muhammad ibn Ibraaheem and Ibn ‘Uthaymeen (may Allaah have mercy on them both) 

Shaykh al-Islam Ibn Taymiyah (may Allaah have mercy on him) said: 

If it is said, “Your wife committed zina” or “She went out of the house” and he gets angry and says, “Then she is divorced,” the divorce does not count. This was stated in a fatwa by Ibn ‘Aqeel, and it is the view of ‘Ata’ ibn Abi Rabaah. Something similar was said by Ibn Abi Moosa. If he said to his wife, “You are divorced because you entered the house,” she is not divorced if she did not enter the house, because he only divorced her for a reason, without which the divorce does not take effect. End quote. 

Al-Fataawa al-Kubra, 5/495 

See: Qawaa’id Ibn Rajab al-Hanbali, p. 323 

Shaykh Muhammad Ibraaheem (may Allaah have mercy on him) said: 

We have received your letter in which you are asking about your divorcing your wife, and you say that your heard something about her so you got angry and divorced her three times. After that you found out that the news was false, and it was proven that it was not true, and you are asking whether this divorce counts as such or not, because it turned out that she is innocent of what was being said about her. 

The answer:  

If the situation is as described, and you only divorced her on the basis of that false news, then the correct scholarly view is that the divorce does not count as such. Based on this, the divorce is invalid and the woman is permissible for you under the initial marriage contract, so there is no need for you to formally take her back or do a new marriage contract. End quote. 

Fataawa Muhamamd ibn Ibraaheem, 11/ question no. 3159 

Shaykh Ibn ‘Uthaymeen said in al-Sharh al-Mumti’, 6/245 

If a person bases his words on a reason which is later found to have no substance, there is no ruling on his words. This is a basic principle which has many implications, among the most important of which is what happens to some people with regard to divorce, where a man says to his wife for example, “If you enter the house of So and so then you are divorced,” based on what that person has of haraam means of entertainment such as musical instruments and the like, then he finds out that he has no such things at all. If she enters that house, is she divorced or not? The answer is that she is not divorced, because that was based on a reason which has no substance. This is based on both sharee’ah and rational thinking. End quote.