Can he delay Hajj because his wife is sick?
What is obligatory for a Muslim when he has the means is to hasten to perform the obligation of Hajj, because the Prophet (blessings and peace of Allah be upon him) said: “Hasten to do Hajj – i.e. the obligatory Hajj – for one of you does not know what is going to happen to him.” Narrated by Ahmad (2721); classed as saheeh by al-Albaani in al-Irwa’ (990). And he (blessings and peace of Allah be upon him) said: “Whoever wants to do Hajj, let him hasten to do it.” Classed as hasan by al-Albaani in Saheeh Abi Dawood (1524).
If the wife will be adversely affected by her husband travelling for the obligatory Hajj, in the sense of real harm and not something that is imaginary, then in this case it is permissible for the husband to delay Hajj until the following year, because Allah, may He be exalted, says (interpretation of the meaning):
“And Hajj (pilgrimage to Makkah) to the House (Ka‘bah) is a duty that mankind owes to Allaah, those who can afford the expenses (for one’s conveyance, provision and residence)”
[Aal ‘Imraan 3:97].
If he fears for his family, then he cannot afford to go for Hajj.
But if the husband can leave with her a woman from among his relatives or a servant to help her, then he has to travel for Hajj, and he may not stay in Makkah for a long time after Hajj.
If he is not able to do that and his wife needs him to be with her, then there is nothing wrong with him delaying Hajj, and he is excused.
And Allah knows best.