He had intercourse with his wife when she was observing a voluntary fast

The one who is observing a voluntary fast has the choice: he may complete the fast or he may break it, but completing it is better. 

Ahmad (26535) narrated from Umm Hani (may Allah be pleased with her) that the Messenger of Allah (peace and blessings of Allah be upon him) entered upon her. He called for a drink and drank some, then he passed it to her and she drank some. She said: O Messenger of Allah, I was fasting. The Messenger of Allah (peace and blessings of Allah be upon him) said: “The one who is observing a voluntary fast has the choice; if he wishes he may fast and if he wishes he may break the fast.” Classed as saheeh by al-Albani in Saheeh al-Jami’ (3854). 

Please see also the answer to question no. 49610. 

If a person fasts one of the six days of Shawwal, and he wants to break the fast, he may do so, whether he does it by eating, having intercourse, or anything else. 

If this woman fasted without her husband’s permission, then he may call her to his bed, and she has to respond. 

But if she was fasting with his permission, then he does not have the right to spoil her fast, but if he wants that then it is better for her to respond to him. 

Ash-Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) said: If she observes a nafil fast with his permission, then it is not permissible for him to spoil her fast, because he gave her permission. But in that case if she is observing a nafil fast with his permission and he wants her to come to his bed, is it better for her to continue fasting and refuse her husband’s request? The other option is better, which is responding to her husband’s request, because responding to her husband’s request is basically obligatory, and the voluntary fast is mustahabb, and because if she refuses even though his desire is strong, then he may feel some resentment towards her and the relationship between them may be affected as a result of that. 

Majmoo’ Fatawa Ibn ‘Uthaymeen (21/174). 

And Allah knows best.